Note that it doesn't matter who paid the tuition. What matters is your dependent status.
Furthermore, there's a new urban myth among college students that says they can get a $1000 from the government just for filing a tax form. For most of them, they simply aren't eligible. A full time unmarried student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on parental support, 529 plans or student loans & grants. You usually must have actually paid tuition, not had it paid by scholarships & grants. It is usually best if the parent claims that credit.
You cannot claim a credit if you are, or can be, claimed as a dependent by someone else.
@hbl3973 is correct. There is a special rule that allows the parent to forgo claiming the student as a dependent; If the student actually has a tax liability he can claim a non-refundable credit but then the parent must forgo claiming the student's $500 other dependent credit.