What is the fate of Moirai & Parcae (the Fates) in Roman & Greek mythologies? Are they mortal? If they are destined to die when & how it happens?
No, the Moirai are immortal. In fact, contrary to to your comment in response to bleh, the gods are also immortal. Henrichs put this most succinctly with the opening of his section on the gods, "First and foremost, Greek gods are immortal (ἀθάνατοι or αἰὲν ἐόντες)", the Greek in which means "being undying or forever". He goes on to say,
"Immortality is the ultimate benchmark of their divinity. Subject to reproduction and birth but exempt from death, the gods worshipped by the Greeks are imperishable but not eternal, unlike the Judaeo-Christian creator god or the supreme being of later Platonists. Differently put, Greek gods have a beginning but no end. Hesiod explains their origins in his Theogony, which embeds the concept of the divine generation in its title. Gods who suffer and die, such as Dionysus Zagreus, are rare exceptions to the Homeric rule that immortality defined as exemption from death is a prerequisite for divinity. Typically, however, dying gods...come back to life and ultimately confirm the principle of divine immortality.
- Albert Henrichs, "What is a Greek God?" in Bremmer & Erskine eds. The Gods of Ancient Greece: Identities and Transformations, p. 31. Edinburgh University Press, 2010.
This was not different for the Romans. The Latin expression used for gods is quite often di immortales (literally 'undying gods'), and in Greek likewise athanatoi theoi (same as the Latin). Furthermore, brotos is Greek for mortal, and it's no coincidence that the Gods drank ambrosia, which lexically also means "immortality".
The mere fact that they're goddesses (which is without question) means they are unable to die.
According to theoi
THE MOIRAI (or Moirae) were the goddesses of fate who personified the inescapable destiny of man. They assinged to every person his or her fate or share in the scheme of things. Their name means "Parts." "Shares" or "Alottted Portions." Zeus Moiragetes, the god of fate, was their leader. Klotho, whose name means "Spinner," spinned the thread of life. Lakhesis, whose name means "Apportioner of Lots"--being derived from a word meaning to receive by lot--, measured the thread of life. Atropos (or Aisa), whose name means "She who cannot be turned," cut the thread of life.
Well, if they were to die, then bad stuff would happen, and fate would end.
Parcae are just the same name, just in Roman form.
Notice that that name means "Parts." If only one was to die, then fate would still probably end.