The image here is from yesterday's Lancaster Sunday News. (I think the paper is now officially called LNP). Generally speaking the paper (using Lancaster as a specific example) offers the parents names, including mom's maiden name, the day of birth and the gender of the newborn. The location, be it hospital or at home, is also provided. Lancaster normally only records the births on Sundays.
Papers are also great for marriage announcements and general articles from time to time. Small local papers often included articles on who was visiting from out of town to local crime to society events.
Often I have found information on a family member simply by just reading through the paper. Granted it is time consuming and sometimes does not yield relevant information, but other times, you can find a gold mine!