It's one single suit with BdB and both city agencies as defendants. Their lawyer probably thought, quite reasonably, that it's simplest and cheapest and therefore best to include all potential defendants where someone could make a plausible nonargument that they should be included for a legitimate non-frivolous reason, even if that argument ends up not winning in court. It prevents DOT or Parks Dept from making the lawsuit slower and more costly by blaming each other and/or BdB and insisting that they be served and joined before resolving the lawsuit.

Also, as a realpolitik practical matter, including BdB in the lawsuit probably helped get the NY Post especially interested in writing about / highlighting her story, which will add pressure on the city to fix the sidewalk faster than otherwise - maybe the extra pressure will get her a quicker and/or better settlement too. The Post quotes her criticizing how BdB ran the city for eight years, and they love to run anti-BdB quotes.