Parayko tied it 1-1 on the power play at 2:35 after Zibanejad whiffed on a clearing attempt.
"I think we're just keeping it simple" said Parayko, who was a game-time decision after missing the morning skate because he was ill. "Just making sure we're getting pucks towards the net and at the net, and that's kind of our biggest thing as a group."
The Rangers outshot the Blues 12-6 in the third period.
"I thought we carried the play for a lot of it, 1-1 game going into the third," New York defenseman Marc Staal said. "[They] kind of get a lucky bounce and we were just unable to find one, but I like our effort. … It also hurt more the position that we're in this time of year to lose a game like that when we played well, but it's going to happen."
Schwartz scored into an empty net with six seconds remaining for the 3-1 final.
The Blues have their second eight-game winning streak this season (Dec. 12-29).
"We don't rely on one or two guys up front or one or two guys on the back end," Schenn said. "We rely on a team to get the job done every night, and on some nights, it's not going to happen. That's just the reality of it, and that's when other guys step up and do a good job filling in for one another, and that's how we're winning hockey games."
