If a parent is claiming a college student as their dependent, do they need to file their taxes before the student?

The parent does not have to file before the student, but make sure the student is filing as a dependent, not an independent return. Otherwise, you won't be able to e-file if you want to claim your student on your return because your student's SSN has been used by him/her.

It is not necessary that one file before the other, but sometimes there needs to be some coordination. A smart move is not to file either return until both returns have been completed.

One common example: Parents have a 529 plan, student has scholarship.  The available  education expenses have to be allocated between the 529 distribution, the scholarship and the tuition credit (if being claimed).

Furthermore,  

There is a tax “loop hole” available. The student reports all his scholarship, up to the amount needed to claim the American Opportunity Credit (AOC), as income on his return. That way, the parents  (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship.  You cannot do this if the school’s billing statement specifically shows the scholarships being applied to tuition or if the conditions of the grant are that it be used to pay for qualified expenses.

Using an example: Student has $10,000 in box 5 of the 1098-T and $8000 in box 1. At first glance he/she has $2000 of taxable income and nobody can claim the American opportunity credit. But if she reports $6000 as income on her return, the parents can claim $4000 of qualified expenses on their return.

Books and computers are also qualifying expenses for the AOC. So, extending the example, the student had another $1000 in expenses for those course materials, paid out of pocket, she would only need to report $5000 of taxable scholarship income, instead of $6000.