West Point has one of the most comprehensive extra-curricular programs of any school. It is well recognized and there, truly is something for everyone. There are military and non-military clubs, there are competitive club sports teams and individual sports teams.
The list can be found here:
http://www.usma.edu/uscc/dca/clubs/clubshome.html
My understanding of the Parachute team is plebes tryout. I think they just had them a week or two ago. You try out as a plebe and stay on the team until graduation. Tryouts are very competitive and you must be in excellent physical condition.
My daughter is on a competitive club athletic team, her sport has a fall and spring season and they practice 4-5 days/ weeks for two hours. She is excused from intramurals. Her team has done some traveling as well.
At the end of Beast, they will have a special evening and present the different clubs and club sports. You can talk to the members and decide what you want to join