Get an opinion from your musician friends... if you think you're really good at piano, then you are. But getting honest opinions from friends will convince you further.
Regarding the parents thing... the best thing to do is ignore them, if you play with headphones, great.. they can't say anything. You're playing some nice and not-too-easy pieces (especially Clair De Lune).. so you seem pretty talented to me.
I don't know how old you are, but stick it out til you can move out. Then you can play piano all you want, or find roommates who appreciate your playing/practicing, or better yet, a girlfriend who loves to hear you play and who appreciates classical and anime piano stuff. Melancholy is GOOD.
Don't ever quit something you feel you're good at. I'm 30 now, and my parents never gave a crap about my music at all, and although I'm not a prodigy at any of the instruments I play (except drums), I feel I'm good enough, and friends online who I show my mp3s to all agree that something is definitely there, and they love my stuff. My parents haven't heard more than 2 songs I've written, and I've written probably 400 songs at this point.
So yeah.... ignore them, be respectful to them, but do your own thing.
By the way, I would love to hear some mp3s of your playing. So let me know if you can send me some!
-Chris