This was the Tigers' first success at Sale since 2018 and their victory in Salford sealed a play-off place with two rounds of the regular season remaining.
"We've got to celebrate this win and our destiny is in our own hands," said Parling, whose side are now just a point behind second-placed Bath, the reigning champions.
"We're now confirmed in the top four and to win here for the first time since 2018, a place where a lot of the players haven't won before, is a big moment, so let's enjoy it," added the former England lock.
Leicester captain Ollie Chessum opened the scoring, but England team-mate Tom Roebuck responded for Sale before Jamie Blamire added a second try for the visitors.
Tigers No 8 Joaquin Moro and Sale flanker Jacques Vermeulen exchanged tries and Joe Heyes added a fourth for the Tigers, who led 28-12 at half-time.
Tries continued to flow in the second half, Will Wand, Ollie Hassell-Collins and Archie van der Flier adding three more for Leicester.
Sale, long out of play-off contention, scored three tries of their own in the final quarter through captain Ernst van Rhyn, Sam Dugdale and Arron Reed to secure a losing bonus point.
"Had the game lasted for another 10 minutes, we could have won that," said Sale supremo Alex Sanderson.
"I'm never happy with a loss, particularly at home, but the lads never threw in the towel. They dug in and there was a brilliant impact from the bench."