Doing presses on a flatter base (palm) gives more stability and control during the press, as well as putting the center of gravity lower to the ground, which also provides more stability. Conversely, this also makes it more difficult strength-wise to do the press because they have to be able to compress their body tightly enough to lift without hitting the ground. It does put a lot of pressure on the wrists, especially during the planche phase, when the gymnast needs to lean forwards to maintain balance during the hip lift. This puts a pretty deep extension on the wrists. This pressure can be alleviated a little bit by slightly turning out the hands so that the thumbs are parallel to each other, but this will slightly diminish finger control for balance. Doing the press on the bridge of the hand, or on the finger tips slightly raises the center of gravity, making it easier to clear the ground during the lift phase. However, it's pretty impractical to do a handstand on the bridge of the hand or fingertips in gymnastics. It's a cool trick, but not really useful. To save the wrists, it's better to do presses on paralettes. It's also wise to do wrist conditioning and stretching as prehab to protect the wrists from injury and other overuse problems.
I once saw "in-ground" paralettes at a gym. There was a raised platform with carpet-bonded foam on top, and there were two pair of paralettes placed under the platform in holes cut to fit them perfectly. The bars of the paralettes came slightly above the level of the carpet, putting them at the perfect height to where it put the gymnast at the exact height they would be as if they were flat palmed. It was a very clever way of saving the wrists during high repetitions, while still giving a "real feel" during the press. I thought it was a brilliant idea! I want something like that at my gym, especially with all the presses we do.
