Latecomers to ball games in the late 19th century were cordoned off into standing-room areas in foul territory.
Because the fans were herded like cattle, this area became known as the “bullpen”, a designation which was later transferred over to the relief pitchers who warmed up there.
I thought it was because the pen was situated near or under the “Bull Durham” tobacco sign. In any event, the children over at peta have no sense of knowledge of baseball history...just their feelings. And, to quote the great Dave Chappelle, “America doesn’t give a f*** about your feelings”