If you're staying off-site, then you are required to pay for parking to visit the theme parks, if you travel to the parks via your own car.
Perhaps your best bet is to stay at a hotel where they run free (included) shuttles to the theme parks.
BoardWalk Resort parking is for guests of the BoardWalk itself, hotel guests or folks patronizing the shops or restaurants on the BoardWalk. It is expressly prohibited for folks to park at the BoardWalk to visit the theme parks, and often a 3 hour limit on parking is imposed just to ensure folks don't do as you suggest.
Although not as agressively prosecuted, it is also such that Downtown Disney parking is for guests patronizing the shops, restaurants and shows at Downtown Disney, and again, not for avoiding the theme park parking fee. Indeed, much of the time (if not all the time) there is no direct transportation between Downtown Disney and the theme parks, presumably because of past abuses.