Were your parents married at the time of your birth? Every state handles it differently. My husband and I weren't married yet when our daughter was born in California in 2005 but fortunately we didn't have to do anything special for the birth certificate as we were both present when the clerk at the hospital took the information. I believe there may have been one extra form to sign though.
In your case, the blood test may have been blood type for determining parentage or it could have been for determining RH status (if mom has negative blood (O- or A- for example) and the baby has positive blood (B+ or AB+ for example) then there are medical consequences. If mom and dad are both negative then the baby will be too and there are no medical consequences to that.
I'd look on the Conn vital records website and, if you can't get any answers from there, call up the state or city/county clerk's office and ask. Post the answer here (I'm curious to see what it means).