I went with "yes, violation" with the understanding that it's one that would be allowed. Perhaps I was wrong to mix "would" and "should".
That's the easiest reading. In a real example, though, there'd be law or authority quoted, and you can be sure that's there would be an exception for emergency vehicles. That doesn't change the meaning of the sign in isolation.
I had my closest park in mind when answering. It has a river and multiple wooden bridges to access. Also, the ground floods sometimes in the winter and some paths are too soft for a car. No way an ambulance or firetruck is going to get in the park. They could try, but would loss more time than doing it on foot, or a bridge would collapse under their weight.
[0] https://99percentinvisible.org/episode/the-wilderness-tool/
You can pretend that's not the case all you want.