On-loan Notts County midfielder Joss Labadie fired Torquay into a third minute lead with his fourth goal since joining the Gulls but Rory McArdle levelled four minutes later, heading home skipper Gary Jones' free kick.
Garry Thompson put the visitors 2-1 up before Thompson crossed for striker James Hanson to head a third.
Ninth placed Bradford are now five points off seventh-placed Exeter with a game in hand.
Parkinson said: "We talked about getting off to a good start and yet found ourselves a goal down after three minutes but we did great after that.
"We got the goal back, a good free kick by Gary (Jones) and a good finish by Rory (McArdle). Then Thommo got the second goal and we were in control.
"The third goal was top class, a great ball in by Thommo and a top finish by James Hanson.
"We could have had more goals and at times we played some good football, despite a very difficult pitch and conditions.
"Torquay have been there or thereabouts for the last two seasons and have won their last two home games so we knew it could be difficult but I thought the players were tremendous.
"We knew it would not always be pretty but like I said the players played the conditions, which we didn't do at Exeter, and we got the result.
"It keeps our season alive and I said to the players that the important thing was getting the three points so that if anyone slips up we are still in there with a shout. It's makes Saturday's game (against Northampton) interesting now."
- Sports & Recreation
- Rory McArdle