Cannot save changes made to Parental Controls profile
The following steps seem to have resolved this issue for me.  (Sadly I don't know if this will help anyone else, since the whole Parental controls system seems very complex and buggy, and I don't really understand what is happening here.  But I thought I'd note it here anyway.)


Delete all unnecessary user accounts (created during testing).
On all remaining accounts, tick "Allow user to administer this computer".  (Here, I'm trying to ensure Parental Controls is disabled on all accounts and hopefully cause some kind of reset, since I'd noticed that a new managed account I'd created had the same issue - and so the issue wasn't with the individual accounts, it was with the entire machine.)
Reboot.
On the account that needs to managed, untick "Allow user to administer this computer".  (I found it immediately turned back on again: some Mac wrestling may be required.)
Quit System Preferences.
Reopen System Preferences and check the "administer this computer" change has stuck.
Press the Show All button in System Preferences.
Cross fingers, wave a chicken in the air, etc.
Open the Parental Controls preference pane.
Set up Parental Controls again.  For me, the settings then stuck and worked.