There is parking at the Beach House, some under the building and some in a parking lot. There is no charge. You must past through a guard gate to get back to the BH which you can do with the car pass you will be given at check in.
Yes there is parking at Beach House & there is also a shuttle that runs throughout day to/from. No charge to park, just keep parking pass displayed that CM gives you at checkin & iirc your keycard is needed because there is gate access there like you find at resort.