Who gets to claim the education credits, the parents or the child?

The education credits are claimed on the return where the student themselves can be claimed (either as a dependent or if they're filing their own return as the taxpayer). This is true regardless of who actually paid for the education costs!

Example: You have a child in college, and you're paying their college expenses. Due to their income, you can't claim them as a dependent on your return. Because of this, they would need to file their own tax return and claim the education credits on their return. You're not eligible to claim the education credits on your return in this case.

Example: You have a child in college, and they're employed and paying for their own college expenses. However, you can claim them as a dependent on your own tax return. You would claim the education credits on your tax return because the child can be claimed as a dependent on your return. This is true even if your child is filing their own tax return. If your child files their own tax return, they'll indicate on their return that they can be claimed as a dependent on your return, and they won't claim the education credits on their tax return.