A few years ago, when the Philadelphia Eagles were in the Super Bowl, my mom's neighbor had a son who managed to score a Super Bowl ticket. Only problem was that after paying for the ticket and the airline fare the kid was out of money. He was only going in for the night so what he did was fly in early that morning, kicked around the city for the day, went to the game and then slept at the airport until his plane left the next day.
Now I would highly suggest that anyone coming to DC come in Monday night because I'm guessing that people will be camping out at the parade route to get a good seat. So if you're planning on coming in just pack light and use the Airport/Train Station/Bus Station until sometime around 4-5am in the morning when you can head out to the parade route (It's January, not sure if you want to camp out there all night). Then just fly home sometime Tuesday evening.
When picking Airports try to get BWI. The MARC has a train that can get you into DC Union Station around 6:30 am. Plus, which airport would you rather fly into for DC? One named after Thurogood Marshall (BWI) or one named after the anti-christ leader of the republican party? (National). Not sure if Dulles has been named after someone.
http://www.mtamaryland.com/services/marc/schedulesSystemMaps/MARC_PENN_HTML_New.cfm