For the second time this season, City had to come from behind but there was no repeat of their dramatic 3-2 victory in October.
Bradford went into this game on the back of four successive clean sheets, but it took Accrington just five minutes to make the breakthrough.
City were unhappy when referee David Webb awarded a free-kick for a foul on Chris Turner and they were even more upset when James Ryan's 25-yard shot found its way past the wall and beyond keeper Rhys Evans.
It was a soft goal to concede and the home fans' anger mounted as the half progressed and Mr Webb waved away a series of increasingly vocal penalty appeals.
The visitors' five-man midfield frustrated City's attacking options and there was little to excite the home crowd apart from the referee's dubious decision making.
Stanley almost grabbed an unlikely second when Robert Grant let fly with a 20-yard strike, but this time Evans got down well to save his goal-bound effort.
Bradford had a penalty appeal for handball waved away just before the break and they came closest to equalising in a late rally.
Stanley keeper Kenny Arthur superbly saved from Nicky Law's 14-yard strike and Paul McLaren saw his follow-up shot deflected over the bar.
In the final attack of the half, winger Omar Daley drilled in a shot from the edge of the box but Stanley centre-half Phil Edwards did well to head it wide.
City kept up the momentum after the break and it was impressive midfielder Law who created the chance for Conlon to level after 51 minutes.
The on-loan Sheffield United man showed a great turn of speed as he burst into the Stanley box and when he drove the ball across the six-yard box, Conlon couldn't miss with a far post finish.
Five minutes later, Accrington skipper Paul Mullin should have done better when he was put through by playmaker John Miles, but the striker hesitated and Matt Clarke got back to make a vital challenge.
There were few clear-cut chances at both sides struggled in the blustery conditions, but City keeper Evans had to be alert in injury time when Miles fired in a shot after more trickery in the Bradford box.
City substitute Chris O'Grady fired into the side-netting in the last attack, but neither side deserved to take all three points.