Your parents are trying to do what they feel is best for you, unfortunately their idea of what is best for you is extraordinarily incorrect. I think, the best way to show them that what they are doing to you is bad, is to tell them your feelings on the matter. Tell them that the furry fandom is in no way a sex fetish, show them evidence of that, show them this site even, Winged Zephyr made a really nice and informative post about what a furry is in the General section. The general ideas is, yes they are ignorant, but that's ok! They only know what a furry is from what people tell them what a furry is and if you tell them what a furry is and give them evidence of that, maybe, hopefully, they will understand.
You already know all the negative consequences of suicide and such, and how it's permanent, and me saying anything might not change your view on that. So I'm going to put this in a spoiler, and it's about why suicide is bad, but sometimes, saying that just makes people feel bored. So read it if you want. If you don't, that's fine.
Spoiler for Hidden:
Problems are truly, only temporary. Any problem you have, or have had, will end eventually and things will get better. To commit suicide you are turning a temporary thing, into a permanent thing, and you are missing out on some great opportunities. Life isn't all bad, is it? Also, this is a decision that will effect everything, to you, once you do this everything you ever know or will, will be meaningless (if you're not religious). You also have to think about people around you, the people who will be effected by your death. It's ok, I know, you're life is really hard and I am really sorry for that, but suicide shouldn't even be an option. It's the worst option you possibly could have in my opinion.
The last thing I'm going to say, is you have a lot of problems at the moment and it's really important, that if you are feeling distressed, to let it out. Venting your problems is a really good way to make yourself feel better, even if it's just temporary. Luckily, this very site has a perfect place for you to vent any problems you have. I think, I really do, you should make a journal (I don't think you have) and post in it when you are feeling really upset. It helps, it really does, trust me.
I hope everything goes ok. These problems will get sorted out, just hang in there!
