Only mozzarella di bufala campana has PDO, even in Italy there are non-PDO, generic mozzarella cheeses produced outside the region.

I see. Similar to Balsamico di Modena / generic balsamic vinegar?

Does non-PDO Parmesan exist in Italy? In the US, "Parmesan" is usually some kind of domestic hard cheese that lacks maturity. The genuine Parmigiano Reggiano costs around 3x.