After a dramatic final run, Parsons finished 0.11 seconds ahead of fourth-placed Martins Dukurs with a combined time of three minutes 22.20 seconds.
Parsons, 30, looked to have lost out on a medal until world champion Dukurs made mistakes on his final run.
South Korea’s Yun Sung-bin won gold by a record 1.63secs and Nikita Tregubov, an Olympic Athlete from Russia, silver.
Briton Jerry Rice, 27, finished 10th in 3:24.24.
“It hasn’t really sunk in yet,” Parsons told BBC Sport. “I thought I’d lost it after that fourth run. It felt like it’d had gone.
“But 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.”
Britain have won a medal every time skeleton has featured at the Winter Olympics, but Parsons is Britain’s first men’s skeleton medal since John Crammond in 1948.
Parsons, who finished 10th in Sochi four years ago and has finished on the podium in the World Cup only once, was not among the main British medal contenders before the Games.
“Dom’s aim was to get a medal. No-one thought that was possible, but he’s loved this track from the moment these Games started,” said Amy Williams, Britain’s Olympic skeleton champion in 2010.
Britons Lizzy Yarnold, the defending champion, and Laura Deas will start their women’s skeleton campaign at 11:20 GMT.