The Gardener
Terrafirma Phobic
I feel fortunate to have had parents involved in aviation and subsequently immersed me in avaition as well. Hell my initials are ETA for a reason! I took my daughter flying shortly after I attained my private certification. Did I subject her to something dangerous? I don't believe so.
I understand people fear certain things in life. Educating them on the aspects of training you undertook to attain your certificate and the preflight standards you have set for yourself are probably a good idea. Then if they finally still don't want their child to go flying with you acknowledge that and move on. Honor their decision. In your case it is really the womans choice since she is 19 but still under the parents guise for financial reasons. Find other people to go with you. Get some cougar to go!
Pffsshhh. Not like you were trained by a 23 year old or anything!
To the OP. Have you given her parents any reasons not to trust you? Do they not let her ride in the car with you or do other 'dangerous' activities? Also, for grins you might offer to take her parents flying first. Did her father ever have any aviation aspirations? It might just be an ego thing too for him. If he wanted to learn to fly and never did he might be angry that you did and he didn't.