Alexander Barabanov had a goal and an assist, and James Reimer made 17 saves for the Sharks (18-32-12), who have lost four straight (0-3-1).
"You can't play a period like that and expect to win a hockey game," San Jose coach David Quinn said. "We went from playing a good brand of hockey to completely abandoning what we did in the first period. Just very sloppy and individualistic."
Barabanov gave the Sharks a 1-0 lead at 3:50 of the first period when he eluded Justin Faulk in the right face-off circle and scored five-hole, and Logan Couture made it 2-0 at 6:31 on a one-timer from Erik Karlsson in the left circle.
Parayko cut it to 2-1 at 2:35 of the second period on a redirection from Nikita Alexandrov, and Kyrou tied it 2-2 at 5:21 on St. Louis' first power play with a wrist shot from the edge of the left circle.
"I thought I saw a good opportunity," Parayko said. "[Alexandrov] had his head up, made a great pass, and [I] just tried to put my stick on the ice in a good spot."
Logan Brown scored his first of the season at 9:30 with a wrist shot from the right circle. It was his 19th NHL game this season.
"I've been getting some chances later, so it was nice to finally put one home," Brown said.