The visitors to Boundary Park took the lead through Nakhi Wells after 21 minutes, only for Jonson Clarke-Harris to pull Oldham level two minutes later.
Bradford, who sit eighth, are now unbeaten in five league games, although only one of those matches has brought victory.
Wells registered his 14th goal of the campaign from close range, having hit the bar six minutes earlier during a spell of concerted Bradford pressure.
Yet Oldham responded swiftly, Clarke-Harris poking home from the edge of the box after good work by James Wesolowski.
Bradford enjoyed the better of proceedings after the break, but were unable to find a winner as Oldham's own unbeaten run was extended to seven in all competitions.
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