Very strange - never seen this before...
What I would try is to uninstall Flash Player completely using the uninstaller from http://helpx.adobe.com/flash-player/kb/uninstall-flash-player-windows.html then reinstall the latest player from
P.S. even if you don't use Firefox, I recommend that you install the NPAPI plugin anyway. It is used by various programs, e.g. Adobe Reader to display PDF docs with Rich Media Content.
Is this someone's work computer that's a member of a Windows domain? If so, it's probably restricted by Group Policy and they'll need to have a conversation with their friendly IT person.
I have run the uninstaller and re-installed several times. I also have both versions of the plugin installed.