For a person to totally skip the gift tax provision, the money must be paid directly from the giver to the school. For example, if the grandparents want to pay for college they can send a tuition payment directly to the school. Even if the annual tuition is $30,000.
Normally, a person can give $14,000 to another person without having to reduce their lifetime exemption for estate tax. Let us say that tuition is $30,000 per year, room and board $15,000 per year, and books $1,000 per year: for a total of $46,000 per year. If they give all the money to the child, they will have to reduce their lifetime exemption by (46K- 14K (Grandparent 1)-14K (grandparent 2)) or $18,000 for that year.
If they send the tuition payment of 30K directly to the school, the remaining 16K can be covered by splitting the gift to the student between the two grandparents.
Now to your situation. The provision you quoted doesn't apply. Sending the payment to the bank isn't the same as sending it to the school. The quoted section is to allow a person to fund another persons education without having the money be commingled with all the other money the student has.
But you can send 14K to each parent, and they can use it to pay off the loan without you having to reduce your lifetime exemption. The good news is that as soon as January rolls around you can send another $14K to each parent. That mean you can send a total of $56,000 to them in the next 6 months. You can double the numbers if you are married and your spouse also send $14K to each of your parents every year.
Unfortunately, the payments are not a source of deduction for you. They are a gift to a family member, they aren't a donation to a charity.