I do not claim my college-age child on my taxes, but I paid part of his college tuition. Can I claim his tuition this year?

If you did not claim the student as a dependent on your tax return then you cannot claim any education expenses for the student on your tax return.

Q.  I paid part of his college tuition. Can I claim his tuition this year?

A. No. The student must be your dependent, in order for you to claim the tuition credit based on his expenses.

But, whomever does claim him, whether it be the other parent or the student himself, they may count the tuition and expenses, you paid, in claiming the credit for themselves. 

Although the general rule, in taxes, is that you must be the one making the payment, to get the deduction or credit, there is an exception for education.