If this is not a transfer, but merely sending the gun to yourself for hunting, for instance, you can ship directly. For a transfer you need not use an FFL on your end, just the receiver. There is no situation where you legally have to use an FFL on the sender's end.
For rifles, USPS and UPS make are pretty easy and cheap. For handguns, you either have to pay for overnight service from Fed Ex, or divide the handgun into parts and ship in two boxes with UPS Ground. I've found the two box method works well when you over insure both halves. $18 total will get the whole thing anywhere in CONUS. You do need to inform them that the box with the frame contains a "firearm receiver", and needs an adult signature, but is not a handgun. The other box is "gun parts" or "parts".
This isn't a trick - UPS recommended this method and I've used it many times.
Just saw the followup. Don't involve a dealer. Send it to yourself care of your parents. State of residence is immaterial as you can do this for hunting trips in Alaska or whatever.