No, that's not the first line of a joke--and the answer is very, very many.
Let's just look at one Saturday morning dual meet. First there are the team reps who work closely with the coaches and who are responsible for most aspects of team management. They may have been in contact earlier in the week, ironing out any details like travel directions or parking or team areas. They've probably also conferred with their coaches and lined up parent meet volunteers.
Friday afternoon, after the teams exchange meet entries, a parent takes the data, merges it on the computer, and prints out all those lane/time sheets and meet programs. Friday night or very early Saturday morning, other parents arrive to set up the pool, moving chairs and tables and getting the meet equipment--backstroke flags, lane lines, starting machine, PA system--in place.
Saturday morning the snack bar crew arrives bright and early to plug in the coffee pot and set out sustenance for the bleary-eyed meet officials beginning to arrive. And there are lots of meet officials--the referee, the starter, four stroke and turn judges, 18 timers (3 parents on each of 6 lanes), a head timer, and an assistant head timer. There are 3 or 4 people at the computer table, 2 or more folks at the awards table, a couple clerks of course, an announcer, and a few runners. There may be a meet manager running around putting out fires--finding the first aid kit, trying to figure out where the pool staff hides the extra toilet paper, calling someone to bring more printer paper.
So, 40 or more moms and dads are busy working so the kids on their teams can compete. And there are even more parents at work at the Wednesday night "B" meets. Many of these parents have done these tasks for years as their child--or children--move through the team from little 8 & under to adult-sized 15-18.
Summer swim team really is a family sport.