The IRS does not say "yes". You're misreading it. Only the person claiming the student as a dependent can claim the tuition credit (or the student himself, in limited circumstances).
But, whomever does claim the credit, whether it be the 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.
