Is the tuition expense deduction for the student's taxes or parents taxes if parents pay the tuition?

Are you talking about the "tuition and fees deduction" or another education benefit? Are you asking about the tuition interest deduction?

The tuition and fees deduction was suspended after December 2020, so it is not available for tax year 2021.

Q. Is the tuition expense deduction for the student's taxes or parents taxes if parents pay the tuition?

A. Neither.

Who pays the tuition does not determine who claims the tuition credit (there is no more tuition deduction).

The person that claims the tuition credit is the person who claims the student's "exemption'' (dependency), regardless of who paid the tuition.

That means that in, most cases, the parent claims the tuition credit.  Only if the student is truly independent, does he/she get to claim the tuition credit. Even then there's a big restriction.  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. 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.