Colleges do not usually include room & board (R&B) in box 1 of the 1098-T. It's a mistake on their part to do so, but it's not unheard of. I'll assume that $3100 is the correct amount of total qualified expenses, but you should verify that.
Q. Who reports 1098-T, parents or student?
A. Both.
You report it to claim the American Opportunity Tax Credit (AOTC) and she enters it to report the taxable scholarship.
It does not matter who paid what. At tax time, you are allowed to allocate educational expenses for the best benefit. 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.
The simple way to this is enter the 1098-T with adjustments (this is allowed, see explanation at "five points" link). The TT interview can get messy and mistakes can be made.
The parent enters the 1098-T with $3100 in box 1 and box 5 blank. The student enters the 1098-T with 0 in box 1 and $5303 (the taxable amount of the scholarship, since there is no longer any tuition to allocate to the scholarship).
