Parking is good for the whole day. You can move between parks for for no additional payment that day.
If you park at the transportation center, one monorail goes to MK and the other goes to Epcot. Epcot also has it's own stand alone parking, as do DHS and AK.
If you're doing Chef Mickey at Contemporary, you can pay same price to valet park there. Eat, ride monorail to MK, return and hand the attendant a few bucks. You need to have an ADR confirmation number, or at least you're supposed to - guard shack attendant just glanced at my computer printout.