I don't think that a woman who has just given birth would be asked to fill out any forms, but yes, she or the baby's father would be asked about their personal info that is needed to complete the BC.
But what I observed was an error in the first two lines which would should contain the same city and the same county for every baby born in that particular hospital, because the hospital remains at the same location. That's what struck me as odd.
LOL Actually, I did fill out forms in hospital
—I remember because I could not decide on a name—I had to make decision prior to leaving hospital, very tense :). I was exhausted too and could not figure out how to spell “Michael”—but then I had just given birth to a 10.5 lb. baby....
Anyway, I did provide the information and complete the forms for my newborn.