Several things, but what you are asking is in the path of spiritual growth. It was similar to my AT thru in some ways.
You need not convince them that the trail is safe, you need to convince them that are not going to stop you from hiking it and living your life as you need to live it, as it is something you need to do. Not doing so would be a death sentence for you (death in a spiritual sense, basically not living the life you were meant to live).
You may in this to convince them that you are capable of this journey, in the sense that many people have done it (and not in the safety sense - again don't go there, don't let them control the conversation that way), but just make the point that even a blind person thru hiked it, and that the trail is very well traveled to the point that everything is laid out to walk.
You may also cut them off on the basis that they need to accept you for who you are, if they do not than they don't know you and you don't care to associate with people who don't accept you for who you are.
Add to this prayer if you can to the degree you can involve it. In this (and using a multi-faith view), you are a child of God, not your parents child, in that pray to God to allow you to go and to open that door and to provide. In this you may find a path to walk, support for you and who you really are, and doors flying open to allow you to go. In this last item, by following this path, I found a new friend, quite out of the blue, 5 months out from my thru hike that drove me down to GA, and picked me up from ME, her offer set the start date. I also got overwhelming support for my AT Thru from about 50 people who I emailed from the trail who sent money, care packages, and paid for stays (and had me stay over with them where practical). I found a overwhelming level of support for who I am and what I was doing, much opposite of my dad who tried to dissuade me (he now supports that I did it).
I also found overwhelming support on the trail.
Good luck