My parents barely believe in mental illnesses, firstly. Basically they technically do believe that mental illness is possible, but that the vast majority are just lazy/over exaggerating/insert excuse of choice. And they wouldn't believe me if I just came to them and said "hey, I have clinical depression." They would need to see it. But I'm barely hiding my symptoms anymore, and they still refuse to notice. They probably have and are ignoring it for whatever reason. And my father specifically is extremely closed-minded and just doesn't get emotion. Well. Emotion that he doesn't share.
And my mother probably has depression herself that she's refusing to acknowledge, not to mention that now her father (my grandfather) died a week ago, and her sister is a real piece of work, so she really just could not possibly handle me having issues like that at this time.
Basically I have thought about this with the help of my best friend for months now, and now at least, I cannot tell my parents.