Since 2008 in the European Union, the word "Parmesan" is defined by law to be only genuine Parmigiano-Reggiano made according to Italian D.O.C. regulations. Therefore, within the European Union, Parmesan and Parmigiano-Reggiano are the same cheese.
In the United States, the word "Parmesan" is not regulated. A cheese labeled as Parmesan in the United States might be genuine Parmigiano-Reggiano, but it's more likely to be an imitation.
Confused? Then ask the chesemonger before you buy. They should be able to tell you if the Parmesan you're buying is the real deal or not.
More About D.O.C. Regulations
D.O.C laws are meant to preserve the integrity of traditional Italian food products by insuring the flavor and quality.
D.O.C. laws require that Parmigiano-Reggiano be made according to a specific recipe and production methods only within the provinces of Parma, Reggio-Emilia, Modena, and specific regions in the provinces of Bologna and Mantua.
So, Does "Imitation" Parmesan Taste Good?
A cheese labeled as Parmesan in the US that is not genuine Parmigiano-Reggiano still can be a tasty cheese. Many artisanal cheesemakers are making high-quality cheeses that are inspired by Parmigiano-Reggiano. Many large cheese producers sell decent Parmesan. Is the flavor as complex as genuine Parmigiano-Reggiano? You be the judge. Buy both and taste them side by side.
Within Italy, cheeses like Parmigiano-Reggiano are called Grana, which means "granular" and refers to a texture well-suited for grating. Many of these cheeses are delicious in their own right. An example is the cheese Grana Padano.