By the park means it's near the park, or close to it Beside the park means it's right next to the park.

Beside is more often used for movable things... For now, this item has been placed beside that.. I like to keep my glass beside the chair... For places, it is common to say next to (which is also often used for movable items.) But I could see saying beside or by for a restroom... For your example, If you can see the washroom from the park, the best choice is to say "the washroom is beside the park. (and then specify what section or attraction of the park the person should find in order to see the washroom.) If you can't see it from the par, use by, Then specify which way from the park a person should travel, and how far.