Do people really not have an our of pocket cost for delivery?

I really want to have birth in a birthing center and was quoted a few thousand for my out of pocket cost. It's about what I expected to pay regardless of if I birthed there or in a hospital, but......

...my dad and step mom swear I shouldn't have any out of pocket cost if I were to choose a hospital birth instead. I don't want money to be the sole deciding factor, but I don't have a lot of it and can't help thinking about how that money could be "better spent" if I did have the option of no out of pocket expense. And the thing is, I actually really like my current OB and would be pretty comfortable delivering in a hospital with him if I had to. But I don't want to compromise my desire for a birthing center delivery only to end up with the same bill in the end. Which, based on the cost calculator on my insurance's website (which I don't know if I'm doing it right though), it's quoting me nearly the same as the birthing center. I do plan to call to see if someone can help me get a more accurate idea, but I just want to know if my dad and step mom are crazy? I've never heard anyone tell me they had no out of pocket cost for delivery and a quick Google search of the average cost is right around what I was quoted by the center and the quote I generated on my insurance for the hospital so.... I think they are crazy.