It is not likely to have any effect on your refund. But if you filed first, it will cause your parent's e-file to be rejected. Or if they filed first and you answered that question incorrectly, your e-file is the one that will be rejected. Duplicate use of SSN's on two tax returns causes rejections. So....who filed first?
If your return is rejected, you can correct it and file again:
Go to MY INFO. Click your name. Go through the screens for the question that asks if someone else can claim you as a dependent. Say YES.
If your incorrect return has been accepted for processing then you have to wait for the IRS to process it fully, receive your refund, and then file an amended return.