Milk does not set on its own. What is used to set it turns out to be the whole story.
Milk does not set on its own
To make cheese you have to split liquid milk into solids and water, and the thing that pulls the trigger is rennet. It is an enzyme from the stomach of a milk-fed calf, which the calf has in order to digest its mother's milk; we borrowed it. Which means cheese uses an animal ingredient beyond the milk and salt — and that is why cheese is not a simple question for vegetarians. Deciding to eat dairy does not settle this one.
For Parmigiano Reggiano the specification settles it
Most cheeses vary from maker to maker. This one does not. The Parmigiano Reggiano PDO product specification states that coagulation is "obtained exclusively by the use of calf rennet" — not a recommendation but a condition of using the name. So Parmigiano Reggiano is not vegetarian, with no exceptions. The same document opens by describing the milk as raw and saying no additives may be used. It is a cheese with remarkably little room to be altered, which is also why this is one of the rare cases where the answer is clean.
The one where a specification settles it:
But 'parmesan' can be a different thing
The names diverge. Parmigiano Reggiano is protected, so only cheese made to the specification may carry it, while parmesan is used far more loosely outside Europe. Some of what is sold as parmesan in the United States or Australia genuinely is made with microbial rennet. For a vegetarian that distinction is the practical one. As a rough rule: the longer and more precise the name, the more likely a specification sits behind it, and the shorter and vaguer the name, the more it needs checking.
There are three substitutes
The first is microbial rennet — enzymes produced by moulds or bacteria, widely used in large-scale cheesemaking. The second is plant-based: in Iberia, milk has been set for a very long time using an infusion of cardoon thistle flowers, and several Portuguese and Spanish sheep's milk cheeses are still made that way. The third takes a micro-organism and gives it the gene to produce the same enzyme a calf would. None of the three passes through an animal, but they are not interchangeable — each cuts the proteins slightly differently, so the texture and finish after ageing come out different too.
Ricotta contains no rennet — and yet
Acid-set cheeses need no rennet at all; the ones you make at home with lemon juice and milk work that way. So this end of the shelf looks safe, except that ricotta has an extra layer. Ricotta is not made from milk but from whey — the clear liquid left over from making another cheese. If that other cheese was set with calf rennet, ricotta inherits the fact without a drop of rennet ever going in. Which means "it was acid-set, so it is fine" does not hold. The thing to check is not how it was made but what came before it.
The one made from whey:
In the end you have to read the packet
The conclusion is that the name of the cheese will not tell you. The same cheese is made differently in different countries, and one company's products may differ from each other. On the ingredients list there are two things to look for: microbial enzymes or vegetable coagulant means no animal was involved, while a bare "rennet" or "coagulant" is best treated as unconfirmed. Some countries label cheese as vegetarian routinely and others never do. When it genuinely matters, there is no substitute for reading the ingredients on the packet in front of you.