I've been wrestling with this for three weeks. Immediately after updating to Yosemite my two kids accounts lost Internet. I'm typically on Ethernet but the problem also occurred with Wifi. Apple senior support had me reset the DNS which temporarily fixed the problem. Along the way we rebooted router and modem. The problem seemed to go away with the router out of the loop, hence the tech's decision to patch me up with a public DNS (8.8.8.8 & 8.8.4.4).
But since the problem didn't affect my admin account it was obviously related to the two managed accounts. I've had issues in the past with Parental Controls, especially with regards to time limits. In this case it turned out for me that turning off the option for "Allow Remote Setup" (in Parental Controls under the little cog when a managed user is selected) seemed to allow normal Internet access again. To verify this I also reset the DNS to my default (the router IP address, 10.0.0.56) and yes, the internet still works for all accounts.
Disabling Parental Controls also worked for me, but obviously this defeats the purpose of having such a thing as Parental Controls in the first place. For reference I have Bedtime limits set (1am to 11pm) and no other limits set - all Apps, Websites, People are allowed. Using latest Chrome, Safari and Firefox browsers. In the course of diagnosis I also deleted the Java applet, but it had no impact.
I hope this helps. I've forwarded this to tech support as well but in the past such things have had no impact. I certainly would have preferred it if Apple acknowledged that an issue that appeared immediately upon upgrading the OS might just be their issue to solve. Blaming the router mfr for an issue that only affects admin accounts and only appeared after the "upgrade" was super shady, IMHO.