Phew, after hours of head-scratching I got the new domains working! What was the problem? An internal CMS redirect.
The nature of the smile site (and my lack of phpwcms experience at the time) meant that I had the default landing page redirect to a new spoof home page that I created to make it appear in the menu structure.
I had entered an absolute path name. Why? I’ve no idea, but by simply changing this to a relative path, the parked domains began to work as a transparent solution.