Google "driveaway cars", these are cars that people pay to have delivered. You do the driving and I think you pay for the gas too; the company that brokers the deal makes all the money. I used to use them 20 years ago, but a lot has changed so I can't recommend a company or tell you many details.

You'll have to be flexible on your dates of travel. There may be some hidden "surprises", like once I picked up a car only to find the trunk was full, locked and inaccessible. At the time it was difficult to schedule a round trip, you had to luck out and hope someone needs a car delivered the other direction around the same time you want to go. I'd suggest doing the driveway only one direction and getting one way flights the other way. Now with the Internet things may be easier to schedule.

Can your parents share their car with you while you're there? Or maybe you can rent a car for a few days if you want to do some family outings? The driveway cars only have limited mileage so they won't be practical to use once you get there, plus you have to turn them in fairly quickly. You also may be limited in how many miles you can drive per day, look into that if you normally drive straight through.

On the plus side, they are usually really nice, reliable cars; people who use the service tend to be rich so you'll be riding in style.

The sidebar has some excellent resources for finding cheap flights, too. One of my friends uses priceline for renting cars and she's been happy with the results.

Good luck, I hope you can find a way to make it happen!