Wow, that's a tough burden to add on top of the already extremely heavy burden of deep depression and suicidal ideation. I've only gone inpatient for care once, and I remember what a tough decision it was to make to do it. I was glad once I was there, because I got much-needed help, but I do remember the mental contortions I put myself through beforehand. And I was an adult with no one to account to besides myself.
I hope I'm not reading things into what you've written, but my impression is that your sense of self-preservation is operational, and that if things get bad enough, that you will do what you need to do to take care of yourself.
The irony is that, if you do go into the hospital, your parents will probably go into supportive "mode".
I don't know if maybe your therapist could help you find a way to communicate to your parents the additional burden they're placing on you with their current attitudes. Maybe
she could tell them.