It depends on the park. At MK and Epcot, you can usually get through security and through the turnstiles by 10-15 minutes before the park opens. You wait there until they do the opening ceremony. In AK, you can go through a little early and walk though the Discovery trails (often they have CMs out with animals for you to see) before hitting the barrier to the rest of the park. In MGM Studios, once you get through security and the turnstiles there are a few shops open before you get to the barrier.
In my experience, at every park you can always get in 10-15 minutes before the park officially opens.