The authenticity of Parmenides has rarely been denied; while Aristotle never refers to it by name, he often attributes to Plato arguments that are either directly from this dialogue or that he heard independently from Plato himself.
You can read Parmenides online in English at the Perseus Project and in French at Wikisource.
This dialogue has a very complicated frame narrative, since the dialogue is narrated by Cephalus, who recounts the meeting of his himself and others with Adeimantus and Glaucon, who took them to their half-brother Antiphon, who recites an account he had from Pythodorus of a discussion involving Parmenides and Socrates.
Cephalus of Clazomenae
Not to be confused with the Cephalus of the Republic. We know nothing about him; but his city, Clazomenae, would have been associated in the Athenian mind with the philosopher Anaxagoras.
Adeimantus of Collytus
Glaucon of Collytus
The half-brother of Plato, Adeimantus, and Glaucon. We know practically nothing about him, but Debra Nails notes that the fact that he spends so much time with his horses suggests that he is a very wealthy man.
He is, in the inner dialogue, hosting the Eleatics, and is the character linking the inner dialogue with the frame narrative. He may be the same Pythodorus mentioned by Thucydides (History3-5). If so, the fact of his banishment may be why his story needs to be received from Antiphon rather than Pythodorus himself.
Zeno of Elea
We know almost nothing about his biography, but he is, of course, the same Zeno who is still famous for his paradoxes. In the context of the dialogue, he has written a book defending Parmenides from opponents arguing that his view is contradictory by arguing that Parmenides' opponents are committed to even more extensive contradictions. The book was published without his consent, but now that it is out, he is going around and reading aloud from it at gatherings. It has been suggested by Ledger that the last part of the dialogue may be based on Zeno's own works, none of which have directly survived. He is, according to the dialogue, almost forty at the time of the discussion.
Parmenides of Elea
Most of what we know about Parmenides' life is quite late, and it is difficult to tell how much of it is embroidery. Fragments of his only known work survive in quotation. John Palmer's SEP article on Parmenides discusses various interpretations of this work. He is, according to the dialogue, roughly sixty-five at the time the discussion takes place.
Aristotle of Thorae
Not to be confused with the Aristotle. The youngest person in attendance at the meeting with the Eleatics, he later became a member of the Thirty Tyrants.
Socrates of Alopece
Socrates is about nineteen or twenty.
In addition there are some anonymous others both in the Cephalus-frame narrative and in the discussion with the Eleatics.
The Plot and The Thought
Cephalus narrates that he and his companions arrived from Clazomenae and ran into Adeimantus and Glaucon in the agora. Cephalus tells them that they are trying to find their half-brother, Antiphon; they have heard that Antiphon knew Pythodorus, who was a friend of Zeno's, and can recite from memory a discussion that took place involving Socrates, Zeno, and Parmenides. They all visit Melite, where Antiphon lives, and Antiphon reluctantly agrees. Zeno and Parmenides were staying with Pythodorus just outside Athens, and Socrates and a number of others visited in order to hear Zeno read from his book.
After hearing Zeno, Socrates begins asking questions to clarify. Zeno takes a hypothesis -- things that exist are many -- and draws contradictions from it. Socrates notes that the basic conclusion of the book is the same as Parmenides', just in different words, and Zeno explains his intent in writing the book. Then Socrates gives his opposed account, that there are forms that make things like other things, and that things are one by participating the form of the One and many by participating the form of the Many. Far from being irritated at Socrates' criticisms, the Eleatics are impressed (although not convinced),and praise him for it. Parmenides then goes on to dismantle Socrates' proposed account in a systematic way. He concludes that, despite Socrates' promise, the reason that Socrates is having difficulties is that he is trying to mark off the forms from each other without having carefully trained himself to do so -- and he recommends that Socrates in fact train himself by doing the kind of thing that most people consider useless wordplay: in other words, exactly the kind of thing Zeno has been doing, taking a hypothesis and following through on its consequences until one hits a contradiction. What is more, he must not be content with only developing one side of the question: "if you want to be trained more thoroughly, you must not only hypothesize, if each thing is, and examine the consequences of that hypothesis; you must also hypothesize, if that same thing is not" (136a).
Socrates notes that this is a massive task, and that he doesn't fully understand the method, so he asks Parmenides to give examples. Parmenides replies that it is indeed a massive task, one for which he is getting a little old; Socrates then asks Zeno, but Zeno agrees that it would make sense for Parmenides to do it. Everyone else agrees, as well, so Parmenides agrees to do it, reluctantly. He decides to carry on a question-and-answer with Aristotle, who is the youngest person present. And the dialogue ends with a very long question-and-answer in which Parmenides and Aristotle simply follow through on the various consequences of a number of hypotheses.
Thoughtwise, the inner dialogue breaks fairly cleanly into two parts, one involving a theory of Forms (separate from the things participating them so as to be like them) and Parmenides' critique of it, and the second given over to the suppositional reasoning of the Deductions about the One (to hen). The Parmenidean Critique has several major arguments:
(1) There seem to be things for which there is no separate Form.
(2) There seems to be no viable account of participation of the Forms.
(3) There seems to be no way that the participating things could be like the Forms they participate. (This is often regarded as the source of the Aristotle's 'Third Man Argument' against the Forms.)
(4) If Socrates tries to avoid these problems by making the Forms mere thoughts in the mind, there doesn't seem to be anything these could be thoughts of.
(5) If Socrates tries to avoid these problems by taking the Forms to be external paradeigmata, models, that participating things participate by modeling, this likeness would itself have to be participation of a Form, ad infinitum, so that participation cannot merely be the likeness of a thing to its model.
(6) There seems to be no way for men to know or do anything with the Forms or for the gods to know or have power over anything except the Forms.
Parmenides' argument is not that the Forms do not exist; everyone agrees that the Forms are required if rational discussion is to have any value. The point he repeatedly makes is that Socrates errs in thinking that he already understands the Forms when he divides one Form from another.
The Deductions are forms of suppositional reasoning, and can be seen as a way of making sure every side of a particular question is understood. The hypotheses discussed are (if I was not too confused by all the questions):
(1) The One is one.
(2) The One is.
(3) The One sometimes is and sometimes is not.
(4) Other things are not the One.
(5) Other things are the One.
(6) The One is not.
(7) The One neither sometimes is nor sometimes is not.
(8) Other things are and the One is not.
All of these lead to various kinds of problems. For instance, it seems the One must be, or it is not the One, but if so, it must participate Being, but then it is not one (being the One + Being). If we try to get out of this by saying that the One does not participate Being, this gives us yet other problems.
Since the dialogue gives no summing-up, but simply ends abruptly after the last Deduction, it's anyone's game to determine what to make of this. But I think it is notable that all of the Deductions seem to make use of this issue of separate Forms that are marked off from other Forms and other things. This is perhaps not surprising -- Parmenides holds that all things are one, and his critique of Socrates focused on both parts of Socrates' account that violated this idea, namely, that the Forms are separate and that the Forms can be easily distinguished from each other.
* The fact that both Parmenides and Antiphon are reluctant to go through the whole of the deductions is surely intentional. Perhaps the implied point is that this training is something best done young.
* It's pretty clear that we are getting an account of how Socrates picks up some of his methods; the examination of everything through questions until he hits a contradiction is, of course, very Socratic, and Parmenides' insistence on talking with the youngest person present is something Socrates himself explicitly does in other dialogues.
* Neoplatonists like Proclus liked to pair Parmenides off with Timaeus as giving a comprehensive account of the principles of philosophy: philosophy's objects of study are the intelligible and the sensible, and Timaeus gives us an account of the sensible while Parmenides gives us an account of the intelligible. Proclus, of course, sees everything as an emanation from the One, so he reads the Deductions as describing successive ways in which divine things emanate from the One, while Timaeus consists of emanations of things in the world from the divine.
Parmenides became the crux of a serious debate in the Renaissance, with Marsilio Ficino on one side argue for a Proclus-style Neoplatonist interpretation, and Pico della Mirandola on the other side arguing that this was wrong. From Pico's work against the Neoplatonists:
I shall say at once, as regards the Parmenides, that in this entire dialogue one does not find a single strict affirmation, and that, in any case, even if there were such an affirmation, nothing would allow one to draw such an inference with certitude. Actually there is nothing less dogmatic than this book, which, taken in its totality, is nothing else than a sort of exercise in dialectic. Indeed, so far are the words of this dialogue from being opposed to my opinion, that all the attempts of critics to read something else into them achieve only arbitrary and willful interpretations.
And this does seem to be the perpetual question: Is this dialogue intended to present something substantive -- whether a criticism or positive position -- or is it intended as an "exercise in dialectic" to guide the reader in the kind of training to which Parmenides exhorts Socrates? Different answers to that question lead along very different roads.
* I tend to avoid complaining about sources, but I have to say, Samuel Rickless's SEP article on Plato's Parmenides is the least illuminating thing I have ever read on Plato.
Quotations from Parmenides are from the translation by Mary Louise Gill and Paul Ryan in Plato, Complete Works, Cooper and Hutchinson, eds., pp. 359-397.