The Wish Park has four zones, each with a portal, an attraction and a bell. To free the captive Pokémon, you must complete the attraction, beat its manager and ring the bell. Once completed, each attraction can be played with other players, up to four. Here are the details, based on Poképédia.

The Wish Park.
AttractionZoneManagerEffect on the Pokémon
Sweets FactorySweets ZoneCofagrigusThey can no longer stop eating
Glamour DanceGlamour ZoneChandelureThey can no longer stop dancing
Furious FightFury ZoneHaxorus, for GothitelleThey become violent
Wondrous FlightWonders ZoneSigilyphThey float uncontrollably

Sweets Factory

The first attraction has two phases: shooting Eggs to score points, then shooting the batter to shape the cake and the icing. The battle against Cofagrigus is also in two parts: destroy its platform of cakes while dodging the Yamask, then face it.

Glamour Dance

The Glamour Dance.

A rhythm challenge: reproduce in time the moves of a Lopunny on stage. The battle against Chandelure takes place to disco music: you have to destroy ten lamps, three speakers and a possessed disco ball, then defeat the ghost while avoiding the Litwick and Lampent it summons.

Furious Fight

The Furious Fight.

Here, the boss comes before the attraction. You first have to run along the top of a wall within the time limit, avoiding the enemies that push the hero back, then beat Haxorus in a duel. The attraction, with a ninja theme, is a reflex game: opponents or bombs pop up in the center of the screen, and you have to either attack or knock the explosive down.

Wondrous Flight

The Wondrous Flight.

A space course of three laps, collecting as many crystals as possible. Then Sigilyph stands at the top of a tower that you have to climb by dodging or eliminating many enemies, before the final duel.

The battle against Darkrai

Once the four zones are freed, the Wish Palace opens. Darkrai is fought there in two phases, first at normal size, then giant. See the walkthrough.

Darkrai.