OK. So using Terminal, in /Volumes I see in addition to . and .. and my HD:
drwxrwxrwx 0 root wheel 0 11 May 11:23 MobileBackups
d--x--x--x+ 3 root admin 102 1 Mar 10:30 UNTITLED
d--x--x--x+ 3 root admin 102 1 Mar 14:32 UNTITLED 1
The MobileBackups is something that Parallels has dumped there. I have removed the UNTITLED and UNTITLED 1 mount points as there was nothing mounted there anyway. After a restart the dialog still appeared (see below). So the dialog is possibly something to do with the MobileBackups from Parallels. I'll dig about in Parallels and see if I can turn that feature off.
Thanks for your help.
I still don't know what is causing this dialog to appear. It is nothing to do with Parallels. The MobileBackups is a Mac OS X thing enabled by default on laptops running Lion and later. It can be disabled with the Terminal command sudo tmutil disablelocal (see also
But disabling the mobile backups does not stop the dialog appearing, so I have no clue as to what this is.