In the song the singer actually says, I ride an old paint and I lead an old dam. A dam is by defintion the mother of a domesicated animal, in this case it's another horse actually a mare. A hoolian is a way to throw a lasso's loop perpendicualar to the ground. This throw is used in catching horses as many horses can see an over-head loop coming at them and will duck out it. The hoolian forms a loop that they run into.