Vernam cancelled out Joshua Kay’s opener but the Bantams are now without a win in four League Two outings.
Barrow were the better side in the early stages and deservedly went in front after 21 minutes when Kay scored with a low shot from outside the box which hit the inside of the post before crossing the line.
Bradford searched for an equaliser and Vernam, who had been close with an earlier effort, twice shot inches wide, the first from an attack down the right and then after goalkeeper Paul Farman had beaten out Oscar Threlkeld’s shot into his path.
Farman charged down a shot from Andy Cook and defender Paudie O’Connor had a long-range effort blocked as the packed Barrow defence continued to thwart the home side.
Having absorbed constant pressure, Barrow almost scored a second goal in stoppage time when a shot from former Bradford loanee Jordan Stevens took a deflection and Richard O’Donnell did well to palm his effort away.
Vernam scored Bradford’s equaliser in the 58th minute, heading Gareth Evans’ right-wing cross into the net at the far post after good work from Threlkeld and Callum Cooke.
Vernam had a late chance to score the winner after racing 40 yards down the middle but fired a yard wide of the far post and Barrow thwarted other Bradford goal attempts to earn a point.