Your situation sounds scarily similar to mine, and while I don't wish to re-tread all these other fine responses, I'd say that yes, you should come out when you feel ready. There is no need at all to tell your parents before the process starts.
I'm in a pretty restrictive situation, having graduated university last year and not had a job since. My parents grew steadily more concerned about this (and still do) while I was toing and froing about my own ->-bleeped-<-. This helped not one jot.
Eventually things slotted into place, and (knowing at last what the issue was) I spoke to my GP back in December on the advice of an NHS video at their Trans Health page. I told my parents that I was seeking help with depression, and that while depression itself was not the problem, I had some things I had to work through. Not a hint of transsexuality.

From speaking to friends in similar circumstances, I gather that the next step can vary, but I was immediately referred to my local mental health practise, just to check for any risks. I'd hoped to be sent for counselling too, as this can be funded by the NHS, but that didn't seem to happen. I imagine that I seemed a little too stable for my own good. >.<
About a month later I finally got to speak to a counsellor, and really it was just to rule out any other conditions, make sure I had home support, and to explain that I would then be referred to Charing Cross for specialist help. It was only after then that I spoke to my parents, and I ended up doing it simply to ease the anguish of forever asking, "shall I talk to Mum today?". I told my Mum when we had the house to ourselves, my sister when I felt more confident shortly afterwards, and my Dad.. a month or two down the line in the middle of a breakdown. That was not so good.
Suffice it to say, things happened when they happened. You will undoubtedly struggle with the urge to tell or hide, but the positive side is that the energy you devote to that worry will help get you through 'coming out'.
Speak to a GP first, though. That'd be my "TL,DR" advice.