Because not to do so could hurt somebody.

Each child under care has the right to privacy. This is to protect the rights of the child. It's also to protect the child from harm from others. As a childcare worker you are never to discuss another child or disclose any information about them to another parent or any other person who comes into the childcare setting.
With the slightest bit of information on or about an adult, criminals can steal their identity. Imagine what freaks and pervs could do with the slightest bit of information on or about our kids.
If someone asks, tell them you could not legally disclose that information. Most parents are fine with this answer, I think, because they soon realize that their own child would be safe under your care because you would not share their information either.

identity theft