Yes, you may claim a tuition credit, even if your parents paid the tuition.  Whether you qualify depends on the circumstances. If your parents don't qualify for the credit because their income is too high, there is a rule against the student claiming the $1000 refundable portion of the credit, even though they don't claim you as a dependent.

 A student, under age 24, is only eligible if he supports himself by working . You cannot be supporting yourself on student loans & grants.   It is usually best if the parent claims the credit rather than the student. He/she does not qualify for the (up to) $1,000 refundable portion of the American Opportunity Credit (AOC) if items 1, 2, and 3 below apply to him. 

You can claim the American Opportunity Credit.  But it will probably be entirely "nonrefundable" meaning that all it can to is reduce your income tax to zero and the largest refund you can get is what was taken for withholding.

I paid for my son's tuition, and he is not my dependent. Can I receive the tuition tax credit on my taxes?

@PropSmith No. But he (or whomever claims him as a dependent) may count what you paid, in claiming the credit for themselves. 

I have claimed my student for 3 years as a dependent. This year my income is above the threshold and wont be claiming her as a dependent. Can she claim the last year of the AOC on her return? She also has an income of $24k and has lived away for more then a year and pays for her own living expenses and I just paid the tuition.

Thanks

@Lipop Yes, your student can report her Education Expenses on her own return to see if she qualifies for the American Opportunity Credit, since she provided over half of her own support. 

Click this link for more info on Education Expenses. 

@Lipop  There are three different issues in your post. 

1. "This year my income is above the threshold and wont be claiming her as a dependent".  

Your income is not too high for her to be your dependent, but is probably too high for you to claim the American Opportunity Credit (AOC). 

2. She also has an income of $24k and has lived away for more then a year.

That means she cannot be your dependent. But that assumes she was really living away.  Being away, from the parent's home,  for the sole purpose of attending school is only a temporary absence and she is still considered as living at home.  In that case her $24K would have to be more than the tuition you paid, for her to be a non dependent.

3.  Even if she is an independent there is another requirement for the AOC. A full time, unmarried student, under age 24, even if you don't qualify as a dependent, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working (earned income). You cannot be supporting yourself on parental support, 529 plans or student loans & grants. 

So they have to die in order to claim the refundable credit? Seems kind of messed up considering that doesn't make a difference on whether you get support or not from your parents.

Yep, orphans get special treatment.