There are always a few blocks in residential areas of DC where they have taken down the signs for road work (or someone has stolen them) and they haven't been replaced. The trick is to identify these areas, park there, and move your car as seldom as possible.
I'm not joking. I did this for months in Adams Morgan.
I'm pretty sure you can also get a visitor/temporary pass at your local Police precinct. Give them a call to find out.