I know you're just looking for a place to park so this may be off the wall, but consider Greenbelt National Park Campground. It's only $16 per night ($8 if you're a senior). No hookups, but is only 13 miles from the White House, just off the beltway. Best park in town. Should be plenty of room this time of year. They have rangers, and the National Park Police also have a HQ there, so it's secure.
Greenbelt Park - Greenbelt Park Campground (U.S. National Park Service)