Luckily I never had this problem with my parents. I joined LPF at the age of 13, around 4 years ago. I did a fair bit of reading, and bought my first 5mW green from eBay. Of course, they were a little curious, however this was long before most of the plane incidents, so they weren't as badly publicized.
I would say, just give it a while. Some peoples parents are more understanding of their child's hobbies (I was playing with high voltage at the age of 10, my father even helped me BUILD it), however some are not as informed about what is dangerous about it and what is not.
Of course, a 100mW green is pretty high to start off with. Like myself and others have said, starting off with a little 5mW green, and using it responsibly (Not burning or lightning matches, not you could do that with 5mW anyway), and maybe doing some safer things like diffraction experiments.
My understanding from your post is that your parents are simply concerned about all the recent aircraft incidents, and do not want you (And in turn, them) to become a part of it, even if you believe you are responsible enough to know not to do it.