We stayed at the Kex hostel in Reykjavik. We parked along the street - don't recall exactly but it was either free or cheap. By the way, we enjoyed the hostel, good beer and food in a friendly somewhat quirky environment. I believe the restaurant area is open to the public. We enjoyed walking around Reykjavik, very interesting architecture. There's a very modern steeple in town, worth the trip up an elevator for the view.