A three-part moon is, apparently, just the moon. As the Wikipedia page about the Triple Goddess that the question links to says, modern Pagan usage such as Wicca considers the moon's waxing, full, and waning phases to be three parts of a single deity, usually identified with Hecate. Another reference to the three-part moon goddess is on the confusingly-named and -written blog called thebible.net, which bills itself as "the most comprehensive collection of non-faith Biblical materials ever assembled":

The Lilim (ללים) were nocturnal spectres, equivalent to the Greek Empusae, and Lilit (Lilith in English) herself the equivalent of Empusa, or Hecate, was their mother; indeed the mother of what we now call witchcraft but in the ancient world was simply the rites and ceremonies of the cult of the three-part moon goddess: new moon (Sleeping Beauty, Red Riding Hood, Cinderella, Maid Marian, Mary Magdalene), full moon (Madonna), waning moon (wicked step-mother, Hecate, Snow Queen).

A poet named Keenan Kelly uses the same phrase in a poem entitled "I'm here where":

I'm now where the three part moon glows
The rise of my love now hides behind the large pine tree
And sadness visits me because you are so far from me
The setting of the moon makes distance sing your name

^{Breathless: A Journal from the Heart. Lincoln, NE: iUniverse, 2005. p. 71.}

Kelly's poem shows that the phrase "three-part moon" is not restricted to neo-pagan contexts any more but is found in literary contexts as a fancy way of saying "moon."

A three-part moon is, apparently, just the moon. As the Wikipedia page about the Triple Goddess that the question links to says, modern Pagan usage such as Wicca considers the moon's waxing, full, and waning phases to be three parts of a single deity, usually identified with Hecate. Another reference to the three-part moon goddess is in a post on the confusingly-named and -written blog called thebible.net, which bills itself as "the most comprehensive collection of non-faith Biblical materials ever assembled":

The Lilim (ללים) were nocturnal spectres, equivalent to the Greek Empusae, and Lilit (Lilith in English) herself the equivalent of Empusa, or Hecate, was their mother; indeed the mother of what we now call witchcraft but in the ancient world was simply the rites and ceremonies of the cult of the three-part moon goddess: new moon (Sleeping Beauty, Red Riding Hood, Cinderella, Maid Marian, Mary Magdalene), full moon (Madonna), waning moon (wicked step-mother, Hecate, Snow Queen).

^{Prashker, David. "Lilit, Lilim." TheBibleNet. thebiblenet.com. Accessed March 23, 2021.}

A poet named Keenan Kelly uses the same phrase in a poem entitled "I'm here where":

I'm now where the three part moon glows
The rise of my love now hides behind the large pine tree
And sadness visits me because you are so far from me
The setting of the moon makes distance sing your name     (ll. 1–4)

^{Breathless: A Journal from the Heart. Lincoln, NE: iUniverse, 2005. p. 71. Google Books. Accessed March 23, 2021.}

Kelly's poem shows that the phrase "three-part moon" is not restricted to neo-pagan contexts any more but is found in literary contexts as a fancy way of saying "moon."