For parental controls to work, the device must use the router’s DNS service. When DNS requests go through the router, the router forces the controlled device to use the DNS services set up in the router. But for devices that are authenticated by your DC, you must give the DC the first chance for DNS resolution, because the router doesn’t have any knowledge of the domain configuration.
If you configure your browser to use the authenticating proxy, then all traffic initiated from the browser will use the DNS service provided by the proxy. This may be different from the DNS service used by the system, i.e., when you authenticate at login.