Instead of parking:<side>=<lane_type> and parking:<side>:access=<access>, I think it would be more intuitive to have parking:<side>=<access> and parking:lane:<side>=<lane_type>.
This seems more consistent with how we tag other access restrictions, and parking:<side>=<access> also allows variations like stopping:<side>=<access> and waiting:<side>=<access>. The side can be omitted entirely if the same restriction applies to both sides, e.g. resulting in a simple stopping=no on major roads (which makes much more sense then parking:stopping=no or parking:both:stopping=no).