only run when someone from the family provides the information for the annoucement. Or in you case, one of the parents should submit the name. Otherwise, just report the number born in the city,county, etc, on a day, week, or whatever statistical basis you want, no names, no identifying information.
I'd ask about birth records. It's long been customary for newspapers to list live births in what is/was called the "Vital Statistics" section of the paper. The information included would be the gender of the child and the name of the parents. More and more, we are seeing either only a mothers name or parents with different last names. In fact, it may be that these two categories outnumber births to married couples. Perhaps the unmarried parents don't care who knows and who does not. I wonder, for those who'd prefer privacy, if publishing this information should be allowed. I know it's used by marketing companies and that's enough to make me think the newspapers should cease the practice.