Q. Can I claim my education credit if my parents already claimed it as well?
A. No. "Double dipping" on tax benefits is not allowed.
Furthermore, you probably couldn't claim it even if they didn't already.
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, 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. You cannot be supporting yourself on parental support, 529 plans or student loans & 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.
Reference: Line 7 instructions for form 8863. https://www.irs.gov/instructions/i8863