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.
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.
- Sports & Recreation
- Nahki Wells
- Greg Tansey