I agree. Although payments direction to a school or medical institution can avoid filing the Gift Tax return, any other gift, either directly or indirectly, would required a Gift Tax return.
To add one more thought to Tom's comment, in the event the OP is married, then each parent could give both you and your spouse separate $14,000 gifts ($15,000 starting in 2018), resulting in a total of $56,000 before a Gift Tax return would be due.
But as Don and Tom said, although a Gift Tax return would be required, no actual tax would be owed.
In the event they want to avoid filing the Gift Tax return, another option would be for them to give you $14,000 (or $28,000 or $56,000, depending on the circumstances) this year, and then continue to give you gifts over the next few years until the mortgage is paid off.
