I know you have already gotten your answers, but, as usual, I always have something to add. Yes, if you stay onsite the parking is free no matter where you go within the WDW resort. If you stay offsite you do have to pay for your first park of the day, but, can hop around all day and use your receipt as a pass for parking at the other parks.
On top of that, I save so much money staying offsite that it more then makes up for the cost of car rental and parking. I can't go to EMH, but, I never went to it even when I stayed onsite. When I stayed onsite I also drove my car and have convinced myself, if only in my mind, that I save a bunch of time and a mega-bunch of frustration by doing so.
To those that feel that they can no longer afford to go to WDW because of the cost of the resort, please try it once offsite. You still get to see the same Mickey, Minnie, Donald and Goofy and you can go home with extra money in your pocket.