The home side were heading for defeat when Terrell Forbes tripped winger Zavon Hines and Connell scored from the spot.
Bradford would have been a couple of goals to the good in the first 11 minutes but for Shots 'keeper Jamie Young, who made two superb diving saves to deny Hines, before then making an equally good stop to tip a 20-yard shot from skipper Gary Jones over the crossbar four minutes before half-time.
However, the visitors, who have now gone six matches unbeaten, shook the home side by taking the lead against the run of play a minute before the interval.
Bradford defender Rory McArdle was booked for a foul on Aldershot's on-loan striker Paul McCallum and the home defence failed to clear Kieron Cadogan's 25-yard free-kick which flashed into the far corner of the net.
Bradford dominated the second half but were denied by a resolute visiting defence until their last-gasp penalty equaliser.