1099-Q has parents name and ss on form but box is checked saying that person not the beneficiary. Who files it?

If it was used to pay qualified education expenses, no one needs to report it, but the expenses it paid cannot also be used for an education credit.

If you claim the student as a dependent and you want to see if you qualify for an education credit, enter the 1099-Q and 1098-T into your TurboTax program. The program will calculate a credit and/or tax on the distribution. 

The tax liability (if there is one) for a distribution is on the person that makes the withdrawal.

The student should always make the withdrawal for education expenses, or distributions can be sent directly to the school. Either of these two ways results in the 1099-Q properly issued to the student.

According to the IRS

“Generally, distributions are tax free if they aren't more than the beneficiary's AQEE for the year. Don't report tax-free distributions (including qualifying rollovers) on your tax return.”