Thanks for your reply. Unlike you, I loathe Word 2011, but it's the only version of MS Word that does everything I need in my work. I looked at upgrading and using Keyboard Maestro and AppleScript to customise it but the programmers have removed some of the underlying code on which my work depends, so later versions simply won't support my work. I am glad that Parallels can allow me to continue to use it, as much as I loathe it, because it does the job (highly specialised scientific editing).
The fact that you installed Mojave as the guest and I have Sierra tells us that the guest OS is not the problem. I put this to the Parallels support man, who observed that Word 2011 likely does something non-standard to draw the screen, which Parallels hasn't taken into account.
It was intriguing to observe that the grey desktop does not appear when either Excel 2011 or PowerPoint 2011 is active. It was also intriguing to see that the grey desktop is present when Word 2011 is visible but not when it is hidden (which supports the tech man's observation).
At least I can still use Word 2011 and continue to earn an income! In the meantime, the programmers are looking into it.