Hi Guilty Dad and
The name of the game might have been helpful for us!
I suspect the disk is looking at the hard drive and sees the install folder, or there is a file that runs when the computer starts and the disk is seeing that it is there and assumes all is good.
When you insert the CD, did you try the uninstall option? That would be my first suggestion. If that works, then you should be able to reinstall it. If that does not work, then you may need to manually delete the game from the Start menu's Startup folder, folders on the disk, and/or maybe even some Registry entries.
Note for security reasons, the children should have a separate computer from the parents - especially if the parent do any on-line shopping or banking or maintain any important work/school/personal business data on the system. And the kids computer(s) should be located in a central location (not the bedroom) where parent can see what they (and any visiting friends) are doing.