If your parents say they will find a way to pay for it, then they must have some ideas where they can come up with that money.
You will get some Subsidized Stafford as well as unsubsidized. Hopefully, you are working right now so you'll have a little bit of a stash yourself, and find a job at the school as well. That could amount to a few thousand.
As to the DIrect Parent Loan, they do have to apply, and now might be a good time to do so. It's a quick credit check and they'll get the answer in a matter of minutes and on the privacy of their computer screen. If one doesn't qualify, the other parent can try. IF they both don't, they can find out why and try to get the reason remedied. They will work with your parents and help them out that way if they are turned down and if it is possible to "fix" the reason. Otherwise, they need a guarantor themselves.
It is deceptive the way some colleges package that Direct Parent loan. It is not an automatic given. The parent(s) have to apply and qualify credit wise. So perhaps, your parents can give it a go right this week so that things can be in place even with glitches in time for you to go to school.