I have a workaround that worked for me - sort of.
Option 1: Create a new user account. This particular parental controls issue seems to only affect upgraded accounts, so a new user account won't have the same issues.
Option 2: Disable parental controls until Apple can provide a real fix or until you can do Option 1. Follow these detailed steps: (a) Promote the affected user to an administrator. (b) Reboot. (c) Log in as that user. (d) Demote that user to a regular user. (e) Do *not* enable parental controls. (f) Reboot. (g) Log in as that user again. The annoying errors should be gone, but you won't have parental controls. If you enable parental controls again, the errors will return, even if you disable parental controls. You'll have to go through the whole promote/demote thing again to make the errors go away.
I don't consider either of these workarounds to be very good, but they may be better than what you're currently experiencing.
Once you fix this issue, you may still have to deal with this issue in OS X 10.8.2 - https://discussions.apple.com/thread/4322178?start=o&tstart=o
I'm also wondering if there's a .plist file somewhere that can just be deleted (and automatically rebuilt at next reboot), but I haven't gotten that far yet.
I found my fix for permanently turning it off. Seems just "turning off parental controls" via "allow unrestricted access to website" under the "Web" pane does not truly turn it off. That's why we all are getting these issues.
I discovered the hidden little "Gear" icon, right above the "Lock" icon. Click it and you find a small menu to "Turn off controls for XX account". Amazing little hidden feature that really should have been implimented better.
It works for me.