Phoenix nursed a 1-0 lead 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, and 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. The Stars won at St. Louis 4-2 Saturday.
Minnesota's win also clinched a playoff berth for the Chicago Blackhawks, the defending Stanley Cup champions.
The Wild won for only the second time in six games and third time in nine games. Phoenix, just back from a three-game trip east, lost for just the third time in eight games.
Just 2:28 into the game, Minnesota was penalized for too many men on the ice. Forty-eight seconds later, Shane Doan's close-range shot deflected off Boedker's leg into the net, the power-play goal giving Phoenix a quick 1-0 lead.
It was Boedker's team-high eighth first-period goal of the season and his 19th overall.
Each team also killed a penalty in the second period.
The Coyotes didn't get a shot from the time Doan missed with 7:49 to play in the second period until Radim Vrbada's with 1:56 left in the game.
Minnesota finally tied it off a face off on a shot by Parise from the top of the circle and it was 1-1 with 11:57 to play.
Spurgeon's powerful slap shot sailed past Greiss high into the net to put the Wild ahead.
Notes: The Wild won the season series 2-1 and would have the tiebreaker. ... The Coyotes are 13-5-0 when Boedker scores a goal. ... Minnesota didn't have a power play until there was 5:17 left in the second period.