Welcome Shaun. Buzzing it down is a good start. You may want to go out and buy a trimmer (like a Wahl or something similar.) Then, you can slowly buzz it shorter and shorter. Start off with the #8, like your parents have said. Then, maybe go a little shorter the next time (like a #6.) Take your time and get comfortable with the short haircut. I think that your parents will get used to it also.
In all honesty, about 90% of the time, we'd all tell you to go ahead and shave it. However, you have some pretty unique circumstances. You gotta do what is best for you, your family, and your safety. So, take your time. You're young and you've got a lot of life ahead of you. There are much worse things in life than losing your hair. So, just get it to the point where you are happy and then go out and have fun. When you're ready, shave it sly and be a proud SBG.
Good luck.
