It doesn't have to be sent anywhere, websites should send a flag when serving various types of content, so parental controls has the information it needs for filtering. Websites could still infer what's being filtered of course, but doing the decision on the client allows for much more granular/extensible categories of content, as well as allowlisting, time limits, etc. Some sites/proxies won't be marked properly, but 99% self-categorizing would help a lot, and it's not like making them do ID checks or attestation is easier than setting an HTTP header or something
Filtering on the app side is more flexible. A game that has some levels not suitable for children could when on a child's device still let them play most of the game.