Lyn has given an accurate response, but I would add that a common alternative is to say "skydiving". "I went skydiving", "I skydived last weekend", "I am going to skydive", "I am going to go skydiving", "I will do a tandem skydive", etc.

Neither of these is correct. You would say "I'm going to do a parachute jump"

Hello Hafsa! The verb for jumping out of a plane with a parachute is "to parachute". For example: I can "parachute". or I enjoy "parachuting" as a hobby. or I'm learning how to "parachute". or I have "parachuted" before." I hope that helps!