Nope, not true. The only mornings when a park opens early is for EMH and those are only for resort guests. Other mornings you will be standing at the turnstiles or at the ropes before the park opens.
Might have been pre-9/11. The only time parks have opened earlier than posted times since are on the 2 or 3 single busiest days of the year. And even then I wouldn't arrive early expecting it to happen--arrive early simply because you gotta.
Some, if not all of the parks will let you through the turnstiles before the posted opening time. But you'll reach a rope barrier before you get to any rides.
They might let you in a little early, but a rope will stop you from getting to the rides. Usually, the rope is right before all the gift shops and a counter service restaurant or two. That way, people can buy their sundries, gifts, or breakfast before hitting the parks.
Went in May, July, and August. Parks weren't open early, but did get in to wait at a rope. Except Magic Kingdom, where we had to wait outside with a bazillion other people.