Is Parmigiano Reggiano vegetarian?

Where the rennet comes in
Rennet is what turns the milk in the copper vat into curd. For Parmigiano Reggiano the disciplinare specifies calf rennet exclusively, alongside raw milk, salt and a natural whey starter from the previous day. That is the entire ingredient list. The step is described in full on how Parmigiano Reggiano is made.
People often ask this together with the lactose question, and the answers go in opposite directions: not vegetarian, but naturally lactose-free. If you are vegetarian and visiting a dairy anyway, it is worth telling the dairy when you book so you are not surprised by the tasting.
Frequently asked questions
Why is Parmigiano Reggiano not vegetarian?
Because it is made with calf rennet, and the production rules allow no alternative.
Is there a vegetarian version of Parmigiano Reggiano?
Not under that name. A cheese made with a different coagulant is not made to the rules, so it cannot be sold as Parmigiano Reggiano.
Is parmesan vegetarian?
Parmigiano Reggiano is not. Parmesan outside the EU is a different, generic cheese; the difference is explained here.