First, the parking lots:
The lots for MK, Epcot, & DHS are always open, because some cast members use the same entrances as the Guest parking. The DAK entrance is not shared by cast parking, and there have been reports by some of not being able to get in in time for their breakfast ADR.
The MK parking lot is actually not at MK, but at TTC. From TTC, you take the monorail to the MK or to CR, Poly, or GF. The Resort Monorail (which also stops at MK) starts running at 7 a.m.
Now for buses:
The buses to the parks start running around 6:30. Disney recommends you be at your resort bus stop 1 hour before your breakfast ADR in a park.
To get to resort breakfasts, you take a bus to the park nearest your breakfast resort, and there change to a bus, boat, or monorail to your final destination. (For Beach Club, you would bus to DHS, not Epcot.) It's best to allow 90 minutes travel time (including waiting), so you might not want to schedule ADR's at resorts before 8 a.m. if you plan to use Disney Transport.