How do I open or become able to enter the Pal Park?

I believe you'll need to obtain the National Dex before you're able to gain access to that part of the game. If I remember correctly, you'll need to see all of the Pokemon available in the Sinnoh region. Then it will be given to you by the professor.

Well first you need to have gotten the national Pokedex, which you only get after you've seen all the Pokemon in Sinnoh. Once you have it you just need to put a GBA game in the slot at the bottom of your DS. (Unless you have a DSi or any of the newer DS game systems. Then you won't be able to use Palpark.)
After this when you turn on your game there will be something that say something like, Migrate from GBA pack. Click on that and pick the Pokemon you want to be transfered to your DS game. Now go to Palpark and catch your Pokemon.

First off, you need to have seen every Pokemon in Sinnoh. Then you talk to Prof. Rowan for the National Pokedex. Next you need a Pokemon game available for the GBA slot. Then go to Route 221 on Platinum. Save the game and turn it off. Put both games in the slots. Turn it on. On the Platinum game, it will say Migrate from the GBA pack. Click on it. It will start to migrate. When it is done, turn it off. Take the GBA game out. Turn it on. You are now able to go in Pal Park.

Beat the Elite 4.