Wells scored both goals in a highly-charged Yorkshire derby, taking his tally for the season to six.
The Bermuda international broke the record when he put Bradford in front a minute before half-time with his eighth goal in consecutive matches - five this season and three in the final three matches of last season.
He collected a headed pass from Garry Thompson and turned the ball past defender Neil Collins before scoring with a low shot into the far corner.
Wells then put the result beyond doubt with Bradford's second goal thee minutes from the end, turning the ball into the net from fellow striker James Hanson's pass after goalkeeper George Long had made a superb save to deny substitute Raffaele De Vita.
Parkin said: "As far as we are concerned he is not leaving Bradford City.
"Other clubs have had 12 months to have a look at him and we are not going to sell him a week before the transfer window closes.
"Both chairmen let us put a squad together to get promoted last season and hopefully to get promoted this season.
"You don't sell your best players if you want success. It is a no-go area.
"It is terrific when you have a goal scorer like Nahki in the team, but he has a long way to go and it is important that he keeps his feet on the ground and keeps learning from the manager and the staff."
Both sides had other chances as play swung from end to end, keeping a bumper 18,042 crowd entertained, but Wells' finishing proved the difference.
United's best first-half chance came after only two minutes when Scot Jamie Murphy outpaced the Bradford defence after running on to a pass from Lyle Taylor, but Jon McLaughlin saved the shot with his legs.
The visitors piled on the pressure in the second half when they forced five corners in a sustained 20-minute spell, the most dangerous moment coming in the 64th minute when substitute Ryan Flynn's 20-yard shot rebounded off Thompson, but McLaughlin turned the ball over the crossbar.
Parkin added: "There was some great defending from our players when Sheffield United had that good spell and to come back at the end and get that second goal was fantastic."
United manager David Weir could not hide his disappointment after seeing his side slip to their second league defeat of the season.
Weir said: "It was a really tight game and we were disappointed to lose it.
"There were chances at both ends, but Bradford managed to get the first goal which was really important.
"A bit of quality enabled them to get that goal and them scoring the first goal made it difficult for us to win the game.
"However, I thought we opened them up and we had opportunities to score. We got into some great positions and kept creating chances and half-chances, but we didn't manage to finish them off.
"There are lots of things to work on and we will improve."
- Sports & Recreation