Unfortuantely websites like these exist.
This site says 246,875. In 9 years, 11 people died on a trampoline. 11 people out of 298,444,215. The chances of you dying in a trampoline accident are about the same as you getting struck by lightning 3 times. This is just the US population.
Approximately 4% of all trampoline emergency-room treated injuries result in hospitalization.
Wow a whole 4%. This is probably an overnight stay because of a broken wrist or ankle.
People like your parents think they are educated on the matter of safety because of media and websites like this. The reality is, if you are smart on a trampoline, you aren't going to get hurt. Yeah you may break a wrist, but that's a risk you're going to have to take. I hate people that are so stuck on making everyone else's lives safer that they forget some people don't like to live in a bubble and eat disinfected tofu every day. Risks are a part of life, and you're going to have to learn that somehow.
Use some parts from this post on your parents and you will be able to get a trampoline.