Park in the sense of tended greenery and park in the sense of stowing your vehicle, though deriving from the same root, diverged in meaning long ago. In Old French, a park was an enclosure. To this dag a military park means an area where vehicles are stored and serviced. As early as 1812 there was a verb "to park," meaning to store one's howitzers in a military park. This carried over to carriages and ultimately to any sort of vehicle.
Our notion of landscaped parks, meanwhile, derives from the medieval practice of enclosing game preserves for the use of the aristocracy. The term was later applied to the grounds around a country estate, then to royal parks in London to which the proles were grudgingly admitted, and finally to any landscaped public grounds. The idea of enclosure is still evident in expressions like "ball park," for an enclosed playing field.
