Is "her" a personal pronoun? I'm not allowed to use them in an essay. How can I change that but still indicate that her parents are buying it for her?
Typically, when writing essays, the only personal pronouns that are discouraged are in the first and second persons.
While in an essay you would be discouraged from saying
I believe her parents bought her a boat.
because of the use of the personal pronoun I, it would be totally fine to say
Her parents bought her a boat.
There is really no reason to have a blanket ban on third person personal pronouns. The only reason I can think of for not wanting to use personal pronouns is to stress the person to whom the personal pronoun is referring, since confusion may arise when saying:
She bought him it because they wanted her to.
But you would want to say
Stacy bought Leonard the boat because her parents wanted her to.