Plenty of skydivers and base jumpers travel (with gear) to go jumping. The TSA might look at you funny* but there is certainly nothing illegal about carrying on a chute.

* Of course everything is always at the discretion of the TSA agent, they can prevent you from getting on a plane because they don't like the shirt you are wearing.

The article seemed more about small airplanes. Big ones have a series of people who would probably be in line to try and fly them before "random passengers with no experience". I don't know anything about planes, but "look for a parachute after you've stabilized the plane" doesn't sound like the worst possible idea, depending on the status of the pilot and other passengers.