Parssinen gave Nashville a 4-1 lead at 17:22 on a redirection from in front on the power play on a pass from Philip Tomasino.
Sharangovich scored on a short-handed breakaway with 45 seconds left for the 4-2 final.
The Predators outshot the Flames 29-13 over the final two periods.
“I really liked our second period,” Nashville coach Andrew Brunette said. “I thought we looked like the team that I’m aspiring for us to be where we track back hard and we counter and play in their zone. To me, [the second] was our best period.”
Calgary visits the Dallas Stars on Friday and the Colorado Avalanche on Saturday.
“I don’t think we had a great start, and then it kind of carried over into the second,” Hanifin said. “I think on both sides of the puck we were a little sloppy and disconnected tonight, which is something we’ve been pretty good at as of late.
“I think this is a game that you’ve got to learn from a little bit, take tomorrow and then get ready for two big games coming up.”
NOTES: Josi had one assist and seven shots on goal in 25:34. He was plus-2. … Predators forward Cody Glass did not play and is week to week because of an upper body injury. … Flames forward Connor Zary did not play because of an undisclosed injury. Forward Adam Ruzicka replaced Zary and had one shot on goal in 6:34. … Calgary forward Jonathan Huberdeau had a three-game point streak end (five points; two goals, three assists). … Flames center Elias Lindholm won 17 of 26 face-offs (65.4 percent).