The Mariners are now just one game short of equalling a club record 22 games without a win, but manager Neil Woods will be pleased with his side's resilience against a Bradford side determined to grab a much-needed victory for caretaker boss Wayne Jacobs.
There were few clear-cut chances for either side in a dour contest, but Grimsby midfielder Peter Sweeney came closest to breaking the deadlock on the hour mark when his shot was well blocked by City stopper Matt Glennon.
The Bantams, who have not won at home in the league since October, huffed and puffed late on without testing Mariners keeper Nick Colgan.
Bradford did come close with three minutes remaining when Michael Flynn's free-kick took a deflection and the ball was inches away from falling into the path of team-mate Matt Clarke.
Source: Team Talk
Source: Team Talk