Parsons went into the last day of the event sitting in fourth position and it appeared that would be the London-born athlete's final placing.
However, Martins Dukurs made errors on his final run, leaving Parsons to dramatically seal third place by a margin of just 0.11 seconds to become Great Britain's first medal winner of the Games.
The 30-year-old told BBC Sport: "It hasn't really sunk in yet. I thought I'd lost it after that fourth run. It felt like it'd had gone.
"Martins Dukurs has made some more mistakes and he's the last person I'd expect that from. It's been great. All the work we've put in has paid off."
South Korea's Yun Sung-bin secured gold by a record margin, Russian-born Olympic athlete Nikita Tregubov finished in second spot while second British representative Jerry Rice ended the event in 10th position.