From the TSA site:


Parachutes

Skydiving rigs with and without Automatic Activation Devices (AAD) are permitted
as a carry-on or as checked luggage.

Typically, a rig will move through the checked luggage or carry-on security
screening process without need for physical inspection. However, TSA screeners
have a duty to thoroughly inspect any item that raises suspicion. If screeners
determine that it is necessary to open a rig for complete inspection, then the
owner of the rig must be present and will be allowed to assist. For this
reason, skydivers are advised to add at least 30 minutes to the airline's
recommended arrival window when traveling with their parachute.

The following recommendations are provided to assist skydivers traveling with
parachutes:

Check the Parachute as Luggage

a.. Pack the rig separately without any other items in the bag. Additional
items, if suspicious, could trigger an inspection of the entire bag.
b.. Screeners will not unpack a parachute without the owner present to provide
assistance. This means that the passenger will be paged and asked to return to
the ticket counter, so they can be present for inspection. Depending on the size
of the airport and passenger volume, it is likely that the checked rig will be
screened within 30 minutes. Remain in the area of the air carrier ticket counter
and pay close attention to airport announcements for up to 30 minutes after
checking in.
c.. If TSA cannot locate the parachute owner, the uncleared parachute will not
be transported on the flight.
d.. Parachute owners may assist TSA screeners to unpack and repack the rig.

Carry the Parachute on the Aircraft

a.. Pack the rig separately without any other items in the bag. Additional
items, if suspicious, could trigger an inspection of the entire bag.
b.. If a further search is required, all efforts will be made to search the
item without out opening the chute(s).
c.. If a chute is opened, the owner can assist. The search may be done in a
location away from the checkpoint to provide adequate space for the search, and
space for the owner to repack the rig.

Parachutists should thoroughly inspect their parachutes at their destination to
ensure that it has not been tampered with or damaged in a manner that renders it
unsafe.