There is no public parking facility for Smithsonian museums on the National Mall. Limited 3-hour metered parking and commercial lots are available—see parking map. Reserved parking near the museums can be purchased in advance through ParkWhiz. Getting Here/Parking: https://www.si.edu/visit/tips
I'd advise taking the subway (Metro). The museum is just a short walk from the Smithsonian station.
Metro is definitely the most convenient. If you must drive look into parking over at L'enfant Plaza
The metro orange, silver, blue line Smithsonian station exit (walk towards National Mall) is across a 2 lane street from the museum. It's sooo easy! Closer than any parking lot nearby.
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