Similar problem regarding the trail head I live near... their response was to put 'No Parking' signs and 'Trail Head' signs that show a pedestrian with a backpack along the side of the road. Signs that people completely ignore. It's 50 mph. And often the problem isn't the lot being full... its that people park on the side of the road because they don't have a discover pass and worry they'll get ticketed.
I don't have an answer... I'm just chiming in to complain.