I did a street view around the area of the park and if I had to guess the reason you can't get an RPP and the reason there are no RPP restrictions in some areas is probably because no one has petitioned for the RPP. It looks like parking for the area across from the park is is actually 2 hour multi space metered as well, is that still in place and does the multi space meter work? Your neighborhood does not seem to look like the type where the developer agrees with the local ANC and future residents can't get parking passes. Usually there are places like Georgetown or west end where existing residents don't want to lose parking. I don't feel that was the case in eckington prior to recent development but I could be wrong.

Can you share the verbatim text in your lease that prohibits getting a parking permit? Honestly I think you may just fill out the petition to zone the block for RPP which will normally allow you to get an RPP. Once you have the petition filled out someone will work with DDOT and at that point can ask that they zone all of the curb space adjacent to the building as RPP not just the side where everybody's Street address is. There are a lot of funny parking signs there - MPD, Fire Lane, Fire Marshall, etc. They look like generic signs that a developer put up for the sake of putting up a sign. Or for selling parking garage spaces.

I don't think you will get much push back for zoning the block for RPP your building contributes aot of curb space to the zone and ther really isn't any reason for other existing neighbors to not want that zoned RPP - unless the goal is parking garage revenue by developer but they have accomplished that goal already it seems.