I poked around a bit but couldn't find anything on your pen either.
As far as I know, Parker never did private label pens. They made a few store branded items but those always carried the Parker name as well.
A more important clue is that your pen is a lever filler. There's no reason Parker would have ever done that in the Vac. era so it's probably not a Parker.
The popularity of the Vacumatics lead innumerable small or regional companies to copy the Vac. styling and materials. Some were quite good quality while most were mediocre to ^&*@.
There's also a chance that you pen was made by a better known UK pen maker. Many made sub-brands for specific markets, to hit particular price points or for a particular distributor or retailer. I'd start there by looking for additional clues such as clip style, section design, lever style or ??
Enjoy the pen and Happy researching!