If the names are considered proper nouns, then they are capitalized.
"Parmigiano-Reggiano" is derived from the names of cities. "Mozzarella" is not.
I would capitalize the first and not the second. Then again, "mozzarella" is a common thing, so it would seem strange to treat it as a proper noun.
In your case, unless you want to investigate the origins of each term, I would consider them all to be proper nouns.
Interested in Language