James Bailey then tried his luck from distance for Barnsley, but his effort whistled wide of the post.
Pickford was again called into action after 16 minutes to deny a long-range strike from Conor Hourihane during a spell of dominance by the home team.
Billy Clarke had a rare effort for Bradford midway through the half, but his attempt was always moving wide of the post.
Moments later, Hourihane stung the palms of Pickford down the other end with a powerful free kick.
Chances continued to arrive as the half progressed, with Rory McArdle missing with a header for Bradford, who had managed to ride out a spell of Barnsley pressure.
Sam Winnall was next to come close for the home team after 36 minutes, but his effort smashed the crossbar as the visitors entered the interval with a one-goal lead.