We are with you, HauntedMansionGeek. If we chose only one park to go to after sleeping late, it would be the ONLY park we would ever get to go to. I've heard that the parks open as early as 9AM, and sometimes 8AM. But as far as we know, this is just a rumor. We do occasionally get out EARLY for an 11:15 character breakfast, but we try not to do this too often. You know, we don't want to wear ourselves out.
You can probably judge by my answer that we don't have any young kids (who seem to want to get up when the sun does) and that we go during off season. You would be right.