The goal, Parise's sixth in his last six games, came halfway through the first period.
The win moves Minnesota (22-20-6) within five points of the Flames and the second wild card spot in the Western Conference. The Wild have picked up five of six possible points so far on a four-game road trip that wraps up Sunday in Vancouver.
Calgary (26-20-3) falls to 1-1 on a six-game homestand, which continues Saturday against the Edmonton Oilers.
Dubynk was at his sharpest in the third period as the Flames generated several dangerous chances.
Six minutes into the third, Johnny Gaudreau got around Jared Spurgeon and fired a dangerous wrist shot labelled for the top corner that Dubnyk stabbed with his glove. With six minutes left in the period, Dubnyk thwarted Paul Byron on a clear breakaway.
Then, as the final couple seconds ticked off the clock, he jabbed out his pad to rob Mikael Backlund in a dangerous scramble at the side of the net.
It was the first time Calgary has been shutout this season after going 71 games since last being blanked.
Minnesota opened the scoring at 8:59, taking advantage of a rare miscue by Flames captain Mark Giordano.
Attempting to send a cross-ice pass from his own corner, Thomas Vanek knocked the puck down and it was immediately picked up by Parise alone in front and he made no mistake, sliding the puck through Jonas Hiller's pads.
Hiller did all he could to keep the Flames in it, robbing Jason Pominville twice in-close on set-ups from Mikko Koivu, the latter chance coming early in the third.
Hiller finished with 31 stops to fall to 14-14-2.
An evenly-played game up until that point, the goal sparked Minnesota, who outshot the home side 8-1 the rest of the period and would have increased its lead if not for Hiller.
Notes: Just called up from Iowa (AHL), Minnesota D Matt Dumba played in his hometown for the first time in his NHL career. His parents and brother were in attendance... Dumba took the spot of Christian Folin, who along with Erik Haula were healthy scratches... Byron has gone 28 games without a goal while LW Brandon Bollig has no goals on the season in 41 games... The Flames have scored 59 third period goals, tied with Tampa Bay for most in the NHL. They've only given up 32.