Parayko scored the Blues' last goal after collecting two assists, and the defending Stanley Cup champions chased Ben Bishop on their way to a dominant 5-1 victory over the Stars on Friday night.
David Perron scored his team-leading 24th goal as St. Louis kept Dallas from pulling even in the West for the third time this week with its third straight win since a 0-3-2 stretch that included an overtime loss at home to the Stars. Jake Allen stopped 18 shots.
St. Louis is four points ahead of Dallas with 20 games to go in a tight race among the top three teams in the West.
Colorado moved into second in the Central Division, a point ahead of the Stars, with a 1-0 win at Anaheim. Dallas has 21 games remaining, and the Avalanche have 22.
"Obviously the standings are close," Parayko said. "However you can get points, it's huge in this league, just try to keep getting points. Not even just Dallas, but any team you can get points against in this league."
Bishop, a 2019 Vezina Trophy finalist, was replaced by Anton Khodobin after Jaden Schwartz beat him from the right circle in transition for a 4-0 lead just 4:45 into the second period.
By then, the drama was already gone in a meeting of teams that played a seven-game playoff series in the second round last season, before the Stars recovered from a horrid start to this season and began climbing steadily in the standings.
Allen ended a personal three-game losing streak while subbing for Jordan Binnington on the second night of a back-to-back.
"We've been building some kind of rivalry against this team and they come in on a back-to-back and we couldn't answer," Dallas defenseman John Klingberg said. "It's disappointing, but use it as a wake-up call and let's move on."
Alexander Steen and Jordan Kyrou also scored for St. Louis, and Robert Thomas matched Parayko with a pair of assists. The Blues improved to 3-0-1 against Dallas, which had a six-game point streak stopped.
Less than six minutes into the game, Steen put in his own rebound off a blocked shot, ending a flurry that began with shots from Parayko and Kyrou.
Late in the first period, Perron easily beat Bishop on a rebound before Kryou and Schwartz, who also had an assist, scored a little more than two minutes apart early in the second. Both goals came in transition, but without traffic in front of Bishop.
Parayko -- who has seven points in four games against Dallas, with four of his seven goals -- made it 5-0 early in the third.
Jamie Oleksiak's third goal and second in as many games with 3:33 remaining kept the Blues from recording a third consecutive shutout.
"It's a good win," coach Craig Berube said. "We knew what was on the line."
The Blues killed four penalties after going ahead 4-0, extending two streaks. St. Louis has killed 12 consecutive penalties over four games, and hasn't allowed a goal on 10 straight Dallas power plays in the past five regular-season meetings.
St. Louis has held all three opponents under 20 shots during its three-game winning streak.
"Guys are coming all the way back," Allen said. "Forwards are breaking up the rush against. They force them to dump pucks in. Really starts with our forwards coming back, defense having good gaps, and we (goalies) have to be there when called upon."
The Blues and Stars have one more scheduled meeting next Saturday in St. Louis. ... The Blues were without D Alex Pietrangelo, who was sick, and C Tyler Bozak, who sustained a lower-body injury in Thursday night's 1-0 win over Arizona. ... It was the Stars' first loss in regulation since Feb. 7 against Minnesota. ... Stars D Roman Polak was given a 10-minute misconduct with 10:44 remaining at the same time he got a two-minute penalty for interference.
Blues: At Minnesota on Sunday.
Stars: Chicago at home Sunday.
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