I was at a barn the night before people were leaving for one of the first big shows of the season. It was chilly, and the aisles were lined with people holding horses with nose chains and twitches while another person pulled the mane. Everyone, including the horse, looked distressed. There is a better way!
The best time to pull a mane is after exercise. The hair follicles and pores are open and the hair will come out easily. Every horse I have ever owned got his/her mane pulled just a bit after every ride, and they grew to associate mane pulling with their after ride carrots and apples. Sometimes I would do a little pulling while they nibbled grass outside. Doing it on a daily, or even weekly basis avoids the night before a show misery. If you have left the chore till the last minute, get a towel, soak it in hot water, and lay it on the mane until cool. This will help!