One thing that can save you a lot of time is to order Dominick's death record. It should plainly state on there his parent's names with exact spelling (hopefully). Once you have the correct spelling, and the mother's last name, you can go from there to find out their parents' names.
Also, if you order ($$) Michele's original Social Security Application, it may contain his exact place of birth and at least his parents' names. It might be harder to find her parents. FamilySearch.org has more Italy births / christenings and you'll have better luck on their site than on here.
I DID find a marriage index for Manhattan (NYC):
Surname, Given Name | Month/Day/Year | County | Cert.#
Angerman, Michele | Aug/26/1911 | Manhattan | #20040
+ Neruina, Caterina
There weren't very many "Michele's" at all - even with a possible transcription error for his last name, and the "U" in her name COULD be a "V" -- so I've been searching on her last name, but no luck. That tells me that it's probably a poor transcription. I did a general search on last name (no exact spellings & no soundex) and came up with these possible spellings: Neirani, Nerino, Neroni, Naroni, Neroni, Nerone, Narrone, Narroni, Nerruno, Nourene, Gnarino, Norena
Tried to track her down in the 1910 Census near where he lived with no luck. I assume she immigrated alone, about age 19/20, and was a boarder somewhere in Manhattan. She may have worked at the same factory as Michele for all we know. :)
Love the story about Dominick and the mob ... my own ancestor left Chicago (1933) during the height of Al Capone's rein over the entire city. Very interesting reading!