I like the respect you show to your parents. Thats OK. Parents like that.
You are 18, it is time you start revolting. Thats OK. Parents want their kids to move, but not too far away
If you happen to live next to a forest (I didnt get if that was a fact or someone making examples, anyway) it is quite amazing that you need to buy a fixed blade knife, they should already be lying around everywhere and so would axes and saws and stuff or did your parents move from the concrete jungle yesterday
Anyway, I do not get the fuss anyway. Go get yourself a nice and decent knife, not a Rambo SERE super kill knife but something stylish, maybe a F1 or some woodlore type knife. A wooden handled knife is more accepted that a knife with black blade. A leather sheath is more accepted than a kydex thigh sheath. A Mora or such are practical but might be seen as just a sharp object and therefore dangerous.
But I am still amazed that you lack access to fixed blades.
Just let your dad in on the whole thing and make him starts fires (he can start with the BBQ) with firesteel and stuff, drag him along to a good outdoors store and look at gadgets and gizmos and then you end up by the knives but there you must be careful. You can not start with the big survival machete, go for some "cute" thing like a small Helle, then your dad might think that it is not macho enough, then you go for the really big knife and that is too big and finally you will end up with the knife you really want in the first place. Not too cute and not too macho. When you finally get home you will have your knife and dad will most likely have some gadgets too. Just dont buy anything from Cold Steel.
Good luck with your knife. Just remember that it is not really your knife until you have cut yourself with it