The original cheese from which parmesan is derived is "Parmigiano Reggiano" -- therefore, American English "par-me-zhan" is actually closer to the original pronunciation than "par-me-zan". The American Heritage Dictionary (which actually uses "par-me-zan") still ascribes it etymologically to Old French "parmigean" = of Parma.

Parmesan, by the way, is not trademarked in the US.