A cartoon image of baseball players sitting under a sign that reads 'Bullpen
The origin of the term “bullpen” has long been debated in baseball. One popular notion is that the term came from Bull Durham tobacco. At one time most ballparks had ads on the outfield fences and Bull Durham was always near the spot where the relief pitchers warmed up. In those days all games were day games and the signs provided much needed shade. In 1910 the Bull Durham name was so closely associated with the ballpark, that signs where in almost every park in the country. These signs stood 40 feet long by 25 feet high. The company offered a $50 reward to any hitter who could hit a ball off one. In addition, any player hitting a home run in a park with a bull on the fence got a carton of tobacco. In 1909 there were 50 signs in place and 14 players won.

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