Phoenix nursed a 1-0 through two periods on Mikkel Boedker's early power-play goal, but the Coyotes went 24 minutes, 28 seconds without a shot.
Parise tied it from the top of the circle off a face off 8:03 into the final period, then came Jared Spurgeon's slap shot from the top of the right circle with 7:03 to play. Parise added an empty-net goal in the final minute.
The Wild pulled three points ahead of Phoenix for the No. 7 spot in the Western Conference. The Coyotes are just a point ahead of Dallas for the No. 8 spot.