Leon Draisaitl got Edmonton on the board early with a even strength goal that came seconds after a power play had expired, taking a nice feed from Ryan Nugent-Hopkins and beating Alex Stalock from near the left post.
Just shy of six minutes later, however, the Wild responded on a power-play marker by Parise, who gobbled up a rebound in front after a Matt Dumba shot, pulled the puck to his forehand and slipped it past Cam Talbot for his fourth of the season.
Alex Stalock was terrific in the period, making 21 saves.
3. Minnesota got its power play going in a big way on Tuesday.
The Wild entered the game among the bottom teams in the NHL in power-play percentage. But one night after scoring once with the man advantage in Vancouver, it connected three times in Edmonton.
Stalock didn't earn an assist on Parise's first-period marker, but give the goaltender credit for leaving his crease and pressing the puck forward behind the Oilers defense. Minnesota forechecked hard, forced a turnover in the offensive zone and seconds later, it was Parise raising his stick in the air.
On Staal's goal in the second, it was a Ryan Suter shot from the point that the savvy veteran centerman was able to get a piece of on its way to the goalmouth.
Staal also assisted on Jared Spurgeon's second-period goal for his second multi-point effort in the past three games and third time this season.
Granlund's winner also came with the extra attacker.