You can bring a parachute on a plane although you most often have to place it with your other baggage and not your carry on baggage.
According to TSA you may transport parachutes, either with or without Automatic Activation Devices, in carry-on or checked bags.
Parachutes should always be packed separately from other baggage.
If a TSA officer determines that a bag must be opened to inspect the parachute, you must be present to assist in the inspection.
IT IS MOST unlikely that a parachute will be of use if a passenger plane crashes.
Even a plane-load of active military parachutists takes several minutes of reasonably steady flight to exit.
Parachuting takes nerve, skill, and strength.
Planes don't provide parachutes to passengers because plane crashes are rare and the plane usually crashes well before passengers have time to get the parachute on and jump.
Also you could still be injured or killed if you jump out of a plane without any experience.