Luton had raced into a 2-0 half-time lead through Asa Hall and Sunday Waisu only to be pegged back by goals from Paul McLaren and Dean Furman after the break.
Hall's 90th-minute header appeared to have won the game for the Hatters only for Conlon's controversial late spot-kick to earn the Bantams a draw and extend their unbeaten run to six games.
Bradford made two changes to the side which drew 1-1 with Accrington with Furman and Peter Thorne recalled in place of Conlon and Michael Boulding, who both dropped to the bench.
The visitors started well with Steve Jones twice going close with long-range efforts inside the first three minutes.
However, it was the hosts who took the lead in the fifth minute when, after Paul Arnison had cut out Ian Henderson's intended pass for Chris Martin, Hall was left unmarked at the near post to glance home Rossi Jarvis' resulting corner.
The match quickly developed into an open affair and chances came at both ends with Law seeing an effort deflected wide in the tenth minute after his initial free-kick had been half-cleared by Wasiu.
Martin then headed over for the hosts in the 21st minute after a cross from the right from Jarvis before McLaren, on his return to his former club, sent a 30-yard effort over the crossbar.
Referee Trevor Kettle made his first booking in the 37th minute when Luke O'Brien brought down Henderson and, from the free-kick, Luton grabbed a second as Wasiu got to Jarvis' cross before goalkeeper Rhys Evans, to tap home from six yards.
Bradford boss Stuart McCall protested fiercely against the free-kick decision and was sent off, electing to take a seat in an empty executive box just yards from the dug-outs.
The visitors almost pulled a goal back on the stroke of half-time when Jones crossed from the left for Thorne to send a glancing 15-yard header just past the post.
The Yorkshiremen kept up the pressure after the interval and just 60 seconds into the second half they pulled a goal back.
Omar Daley's 20-yard shot was deflected wide allowing Jones to send in a corner, which caused a melee in the box before being tucked home by McLaren from close range.
The momentum was with the Bantams and in the 57th minute they were level as Jones and Law combined to find Furman, who arrowed in a superb effort, left-footed from 16 yards for a fine equaliser.
Daley then ended a scintillating 70-yard run with a 20-yard shot over the bar on the hour.
Luton awoke from their slumber in the 71st minute as Kevin Gallen sent Waisu running clean through only for Evans to save at the striker's feet before Martin curled the rebound just wide.
Hall rose to head what appeared to be the winner in the final minute after a curling free-kick from substitute Kevin Nicholls but Conlon's late penalty saved the day for Bradford, while Martin was sent off for Luton in the arguments that ensued.