Greg Tansey's spot-kick had originally put the hosts in front before Reid's picture-perfect response, while Nahki Wells' record-breaking run of scoring in eight consecutive games finally came to an end.
After a slow start, the game burst into life midway through the first half when Luke Freeman's shot was blocked in the penalty area by Gary Jones' arm.
Referee Gary Sutton pointed to the spot and Tansey stepped up to send keeper Jon McLaughlin the wrong way and give Stevenage the lead.
The hosts almost doubled their advantage 10 minutes later when James Dunne's shot was cleared off the line by Nathan Doyle amid appeals for another handball from the home side.
Bradford went straight up the other end and almost levelled but Wells could only fire over on the stretch from Reid's cross.
But the visitors did not have to wait long to equalise as Reid curled home a stunning free-kick into the top corner two minutes after the break.
Substitute Oumare Tounkara almost nicked it for Stevenage late on but his effort was brilliantly tipped wide by McLaughlin.