Wmata (Metro's operating authority), has been doing extensive track work on the weekends. My guess is the work will not happen the weekend after the 4th (which, unfortunately, is a Thursday this year) due to the tourist crush. Red Line, especially north of the RI Ave stop, has been under constant reconstruction (with some stations closed on weekends, buses replacing trains). Greenbelt would be best because, as was mentioned, there is plenty of parking. Just be aware that parking may still be at a premium on the 4th. Trains will also still be packed. I rarely park and ride (unless visiting family in Mt. Airy, driving to Shady Grove, and riding into DC), so someone with more experience, especially with park and ride on the 4th, could probably give the best answer.
Wmata's current work calendar.
Metro - Rail - Major Weekend Track Work
I was wrong, there is Orange Line work on the weekend after the 4th. Too get to Smithsonian, take Green Line towards Branch Ave, then exit at L'Enfant Plaza. Take Orange (Vienna) or Blue (Springfield-Franconia) one stop to Smithsonian.
Good luck!