The only way to move Pokémon from Pearl to Diamond and vice versa is by trading; the Pal Park, PokeTransfer, and similar facilities are only for moving Pokémon from one generation to another.
A generation is a group of games classified by the total number of Pokémon that exist during the games release, with each generation introducing roughly 100 new Pokémon each. So for example, a Gen 3 game will have just under 400 Pokémon, while a Gen 4 game (like D&P) will have just under 500 Pokémon. Every game since Gen 4 has also included a method for one-way transfers of Pokémon from the previous generation (like Pal Park, PokeTransfer, and PokeBank), a feature that doesn’t apply for games of the same or higher generation.
These games can’t connect to PokeBank directly, so unfortunately, you are going to need two systems in order to move Pokémon. If you have any friends that also play, I would recommend asking to borrow their system if possible (and it doesn’t matter if it’s a regular DS, 3DS or 2DS; if it can play the game, it can communicate and trade with the other game).