Parmesan has become the type name, and Parmigiano Reggiano is the protected name for the Italian cheese. There’s actually an urban legend that the Parmesan made in Wisconsin is the more authentic, historically-accurate recipe, because the cheese made in Parma and Reggio Emilia has evolved over the years, while the immigrants in the United States stayed true to the old methods.
Parmesan has become the type name, and Parmigiano Reggiano is the protected name for the Italian cheese. There’s actually an urban legend that the Parmesan made in Wisconsin is the more authentic, historically-accurate recipe, because the cheese made in Parma and Reggio Emilia has evolved over the years, while the immigrants in the United States stayed true to the old methods.
Slate has a long read on the topic.
TL;DR:
Neither one is “authentic.” Both Wisconsin Parmesan and Parmiggiano Reggiano have evolved independently.