My uncle- who married an Italian - called it par-MAY-zen. Dialectical difference, maybe.
Accounting for inaccuracies in reproduction and perception, [par me zan] is a plausible Italian pronunciation of the English word “parmesan,” which is different from the Italian word “parmigiano.”
It’s also plausible for the American spouse of an Italian to render [par me zan] as [par ˈmeɪ zən], which would account for the uncle’s pronunciation without any need for speculating about an Italian dialectical pronunciation.