As an airplane junkie and aeronautical engineering student I can tell you I had a similar problem with my parents, although not that severe. Since I taught them what happens in the cockpit during the flight, what are the physics involved in flying and piloting basics including navigation procedures, etc. etc. they fear has disappeared. I have to admit that flying is pretty much unnatural and it doesn’t feel good when you don’t know what is happening / what are you feeling, but once you learn what’s behind scenes the confidence is extreme. F.e. my mother was extremelly afraid of that sensation of free fall you might have during a flight but when I explained her why that happened (vertical speed decrease or sudden downwards airflow pushing the aircraft as a whole) she started to lose panic. It then became bereable.