So, there are a couple things here. _Parcae_ refers to the Roman Fates, while Moirai (or Moerae) refer to the Greek Fates.
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.