No parking at Renwick Gallery as it is right by the White House but you will find some street parking a few blocks away and some underground parking in buildings not too far from there.
It is better to use public transportation in DC, including taxis, if you are doing the tourist thing. I drove to DC, left my car in my hotel's valet parking for 5 days and used Metro for the entire visit. Plan your visits along the routes. BTW, This will be my fourth trip in two years, and I've still not seen enough of Washington.
It's just better to use the Subway system
No public parking for the gallery. It is on the stretch of Pennsylvania Avenue that is The Pedestrian Plaza for the White House North Front. There are meters on the streets and commercial parking garages in the neighborhood but they are expensive.
Street parking only. Too close to the White House. Parking structures are nearby.
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