Where I live, therapists are bound to confidentiality. The only time they're allowed to tell someone else what you've told them is if it involves hurting yourself or someone else, so talking about suicidal ideation can be iffy. I think self harm falls more into confidentiality, since your life isn't actively in danger. When I've spoken about self harm to my therapist, I didn't get any calls from home or anything like that. Beyond that, it would be illegal for your therapist to tell your parents anything.
For recovery, I think you should give it a chance. I have a lot of trouble wrapping my head around the idea of a version of myself that is 'better' and no longer depressed. If you've been struggling for a long time, then progress will be slow, and sometimes it'll get worse before it gets better. Through therapy and medication, I can say that my mood's at least improved, on average, but there's still a whole host of deeply rooted issues that I'm going to have to work out.
I can definitely relate to thinking that it won't work anyways; my thinking tends to be pretty black and white when it comes to these kinds of things. But you might as well try, right? If it turns out that there's light at the end of the tunnel, then it can be really worth it. It might take a long time before that light ever shows itself, though.
It's great that you're seeking recovery. Definitely give it a chance, chances, many chances, etc. You'll always have the option to CTB, so recovery can really only be a good thing. Things can start to get better, but if that never happens, then at least you've really tried. That's all anyone could really ask for.
