How is your relationship with your parents? If you have a good relationship with them and think they would be understanding, and maybe even supportive, then you could tell them. do they not take well with mental illness because they don't understand it? Are they teachable? It's really up to you, and how well you know them....my parents have been very adaptable. My husband, who really knew nothing about mental illness until he met me and also "didn't take well with it" is also handling it pretty well. Take it slow...maybe mention 'a friend" to start with...talk to your counselor about how to approach it maybe?
Good luck - it's tough to handle on your own but definitely not impossible, and you have us here at PC to help you through it.