The buses start running by 6:30, but they don't run quite as frequently as the regular resort to park buses run. About an hour or so before a park opens, the buses start running more often. Let's say MK opens at 9am, the buses will start running every 10-15 mins by about 7:45-8am. If the MK has EMHs that morning, the buses will start an hour before EMHs start...so by 7ish.
You are free to take any bus to the parks. I wouldn't recommend taking a 6:30am bus to MK if you just wanted to wait for the park to open though. No reason to do that. An hour before the park opens is early enough.