I'm fascinated by the fact that my takeaway is the precise opposite of what the author intended. To me, the answer to all of the questions was crystal-clear. Yes, you can academically wonder whether an orbiting space station is a vehicle and whether it's in the park, but the obvious intent of the sign couldn't be clearer. Cars/trucks/motorcycles aren't allowed, and obviously police and ambulances (and fire trucks) do…
They basically said to ignore local jurisdiction laws. Depending on where the sign is placed, this might not be so obvious.
If you take the sign rule to the letter, an ambulance is a vehicle and thus cannot enter the park as well.
Police can enter the park, but without a vehicle.