I could totally get behind Western martial arts. Fencing has become a gross parody of real swordfighting and could honestly be dispensed with.
Parkour is expressly non-competitive. One of the few things the divergent tricking vs. utility factions agree on is that you help people progress, not try to prove you’re better than them.
Diving rules would have to be changed. Right now, there’s a Degree of Difficulty (DD) multiplier added to the score. The most difficult dives involve, obviously, more flips and twists, and the top men again outperform the top women.
Rock climbing, again, nope. The top men outperform the top women, though women are far more competitive against men than they are in most other sports.
My votes:
[ul]
[li]Paintball[/li][li]MMA or the original version, Pankration[/li][li]Roller derby[/li][li]Archery biathlon[/li][li]Strongman competition[/li][/ul]