There is no public parking facility at the Smithsonian on the National Mall; on-street parking is limited and posted times are enforced.
There are commercial parking lots and garages located within several blocks of the museums. On Madison Drive there are three parking spaces at the East end of the Museum and two parking spaces in front of the Museum for people with disabilities. These are standard curb spaces on the left side of a one-way street.
The closest parking garage to the National Museum of Natural History with accessible parking spaces is in the Ronald Reagan Building (14th Street and Pennsylvania Avenue, NW)