The 30-year-old came in third after a tense race in which he thought he had blown it but Latvian Martins Dukurs lost time on the lower part of the course, leaving Parsons to take bronze.
It is the first time Team GB's men's skeleton team has won a medal since 1948.
The skeleton race became highly contentious after rival teams questioned the legality of Team GB's skinsuits.
Parsons insisted sliding techniques have a greater influence than the suits.
The racer, who came 10th in the 2014 Sochi Games, said his improvement was down to working with former British number one Kristan Bromley, whose brother Richard coaches the South Korean sliders.
The International Bobsleigh and Skeleton Federation has approved the British skinsuits for the competition.
