No Pokepark has a story to it you need to pass the minigames and challenges to continue the story.
One day, the legendary Pokémon Mew tells Pikachu, Charmander, Chikorita and Piplup that the 14 pieces of the Sky Prism have gone missing. Pikachu and friends set out on an adventure to find them in the PokéPark, which encompasses various zones such as the meadow and lava zones. Pikachu makes friends with other Pokémon and can use their abilities to clear various attractions in the PokéPark. By clearing attractions, the player will obtain pieces of the Sky Prism.
You go around Pokepark to finds these pieces and prevent it from falling. After the story there are a few challenges but you didn't ask about that xP