My girlfriend and I have both traveled with parahutes. She has even
taken it as carry on baggage. No problems so far.
I have a laminated tag on my chute bag:

Airport Security:

(TSA logo)

This is an emergency parachute

Ø It has been inspected and repacked by an FAA certified rigger.

Ø My life may depend on this safety equipment. Please respect it.

Ø If the seal is broken, or it appears to have been tampered with, it
is not considered airworthy.

Ø X-ray it and swipe the container all you want.

Ø Questions: You know where to find me.

Thanks!