The next year with nearly 150 Bull Durham signs being hit 85 times, $4,520 in cash and more than 10,000 pounds of tobacco was given out. While this seems to be the most convincing theory of the terms origin, it should be noted that the term “bullpen” had long been used in the United States to denote either a log enclosure for holding cattle or a holding area for prisoners. This concept of it being an enclosure, along with some help from the Bull Durham promotion, may have strongly influenced the terms use.
Another theory likens the relief pitchers to the reserve bulls in bullfighting, who are pinned nearby the arena should the starting bull be deemed unable to fight.
Reference-The New Dickson Baseball Dictionary, Paul Dickson
Also read Origin of 7th Inning Stretch