Not had a problem either carry on or checked parachute.
Taking a parachute as checked baggage, mine always has a laminated tag on the carrying bag, as below.
Something that has raised more than a few eyebrows is the Handpresso handheld espresso machine I carry on. Like a small bicycle pump with a bit sticking out the side at one end, it looks a bit like "one of those".
Jim

Airport Security:
(TSA logo)
This is an emergency parachute
It has been inspected and repacked by an FAA certified rigger.
My life may depend on this safety equipment. Please respect it.
If the seal is broken, or it appears to have been tampered with, it is not considered airworthy.
X-ray it and swipe the container all you want.
Questions: You know where to find me.
Thanks!