One solution is to form relationships with other organizations with parking. When I was attending a church in Arlington, pre-Whole Foods. Yes, that Whole Foods wasn't always there. Anyway, there used to be an empty lot across the street from the church. That was fine until someone decided to throw a building on top of the lot. So what the church did was form a relationship with a nearby bank that was closed on Sunday, and get parking passes for parishioners who needed them. These passes would be highlighted in the church bulletin and mentioned during announcements. Church members were told if they park illegally they would get ticketed or towed, this was also mentioned during announcements. Come to think there are many churches I've visited that have such an arrangement, where they seek out extra parking from businesses and non-profits that have parking spaces on Sundays. My neighbor mentioned a church that formed a real relationship with the neighborhood, so that neighbors allowed church members to park in their driveways. No such relationship seems to exist here in Shaw, so moving on.
Once I was visiting St. Johns Episcopal near the White House. They have valet parking. I have no idea where the valet goes when they park a car when we go out to a restaurant, so I have no clue where church valets park either.
Another solution, get your own friggin parking lot. There are several churches and worship centers with some parking. The mosque on 4th St/Islam Way has a parking lot. Scripture Cathedral has a parking lot. But then again so does Shiloh Baptist, but there are more member's cars than there are legal spots.
A more drastic solution is move to where your congregation lives. The Help and I have been working on the archives of his church and his church started off in the early 20th century in DC. Then when the congregation started moving to the burbs, the church moved with them. In the 21st century they moved again and built a new church, selling their old church building. The congregation continues to grow and so the church plants daughter churches. I have a friend who belongs to a tiny Antioch Orthodox church. It is only tiny because it only wants to be a certain size, when the congregation gets to a certain size it also plants daughter churches to maintain its smallness, which is more suited to its character. It would not be the same church if it were bigger. Regarding parking, many congregants are encouraged to live within a close distance in order to participate fully in church life, so no car is needed for services. Then there is the old idea of going to the church that is closest to you, the church in the parish where you live.
Sadly too many Christians mistake a church building for the real meaning of church. A church is the body of Christ, in the people of Christ, not in the building the people gather in. Lastly, the roads are government's, or the District's. The DC government maintains them, cleans them, repairs them and has the right to meter them and charge for using them, so remember Matthew 22:20-22.