Learn French: Ils auraient étudié la médecine par vocation. - They would have studied medicine as a vocation.

The past conditional is a compound tense. You form it by taking the conditional present of the auxiliary verb (in this case, avoir, which becomes auraient for ils) and adding the past participle of the main verb (étudié).

Why is there a la before médecine? The English translation doesn't say "the medicine".

In French, you must use a definite article (le, la, les) before general concepts, broad categories, and fields of study. Since medicine is a field of study here, it requires the feminine article la.

Why use par in par vocation instead of another word like comme (as)?

While the English translation says "as a vocation", the French preposition par is frequently used to express a motive, reason, or driving force. In this context, it means they studied it "out of a calling" or "motivated by vocation".

If the subject was feminine, like Elles, would étudié need an extra "e" for agreement?

No, it would remain exactly the same: Elles auraient étudié. Because the auxiliary verb is avoir, the past participle étudié does not agree with the subject. Agreement with avoir only happens if a direct object comes before the verb, which is not the case in this sentence.