How can I change my child's birth date?

You can't change the birth date of a child account own your own, you will need to contact apple:

Update the birth date associated with your Apple ID - Apple Support

"If you need to change the date of birth for a child, contact Apple Support. "

Contact apple with that link or:

Contact Apple for support and service - Apple Support

Good luck!

Note the under 13 years old restriction in the article below.

Apple ID - Update the birth date associated with.         

If necessary, contact Apple ID Support for assistance.  See the Apple Support articles below for contact information.  

Apple ID - Official Apple Support.    

Contact Apple for help with Apple ID account security - Apple Support.   (Link doesn’t always load)  

Get Support          Work through the screens to get a chat option. You may have to pick a slightly different problem.

Apple Support schedule phone call

https://twitter.com/AppleSupport.          People have posted it may take an hour or more to get a response.