Parrish scored on a backhand shot that fooled Ducks goaltender Michael Wall with 4:18 left in the third to send the Wild to their NHL-best 16th home win. Brian Rolston also scored for the Wild.
Ducks forward Teemu Selanne, playing his 1,000th NHL game, scored two goals and added an assist as the Ducks rallied to tie the game after trailing 3-1. Anaheim also got a goal from Scott Niedermayer.
Wall, playing in just his fourth NHL game, made 25 saves, while Manny Fernandez had 27 saves for Minnesota.