Who files the 1098-T form? The parent or the student?

If the student is being claimed as a dependent on the parent's tax return, the Form 1098-T is entered on the parent's return.

Simple answer: the parent.  Even if the student paid the tuition. The tuition credit  was created with the parent, in mind.  There are restrictions on a college age student trying to claim the credit on his own return.

There's even a "loop hole" available for the parents to claim a tuition credit if the student is on scholarship. 

The 1098-T is only an informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or possibly your student has taxable scholarship income (in which case the student files it). So, there are times when both the parent and the student "file" the 1098-T.  

If you claim the tuition credit, you do need to report that you got one or that you qualify for an exception (the TurboTax interview will handle this)