According to theoi.com, Parcae/Moirai means "parts":
Their name means "Parts." "Shares" or "Alottted Portions."
Is there a symbolism behind this? Is it because Fate was split into 3?
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Next, "parts" is a not particularly helpful translation. The better one is the third one you shared, "allotted portions" (or apportioned lots). It instead refers to their activity. There were three Fates in Greek mythology, which you can read if you just look a little further in the same paragraph you link to:
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.
The "parts" or "apportioned lots" refers to their activity. The spin, weave, and cut the "thread of life", so to speak. The life you get is the life they allot you.
Of course, the deeper side of this is the worldview it reflects: a cosmic determinism and a mechanistic precision. Even Zeus/Jupiter had to bow before the Fates. An individual's lot was predetermined at birth; the process was in most cases impersonal; and when one's exact time was up, there was little negotiable latitude. There may also be significance in the 'sacred number' 3: female gods often come in threes (e.g., Graiae, Graces). In Indo-European culture, the female is often observed mediating binary oppositions, like life and death.