Health Insurance Questions

I'm under my parents' health insurance and a few of their other insurances. I'm pregnant and 20 and I'm planning on moving out to live with my baby daddy. Problem is, My parents don't know I'm pregnant. & in order to keep this a secret... I need be covered by my own insurance so she doesn't receive my doctor visit bills & so I can save $... So my question is, Is it possible I can still stay under my parents insurance yet have my own? Like Obama Care? Or do I need to be completely off their insurance. Also one other question, let's say my baby daddy qualifies for his own insurance (he doesn't have any yet) can I be filed under his insurance as well or would we have to be married in order for that to work? I'm going to call an insurance agent anyway but just thought I'd ask anyone who knows here