As a retired Museum Curator, establishing the history or provenance of an artifact such as a firearm can be challenging, rewarding, and frustrating.   Like many have stated it all begins with a serial number.   Little else can be done without it.   Even then, the history may simply end with the original shipping information from a manufacturer.   With that said, a couple of thoughts: 
 
You mention that it was ordered with no engraving.   You also state that you recall a plain, unmarked, grip cap and butt plate.   Additionally, you have not sighted or heard of the gun in decades.   It is possible that it may not have been a Parker. 
 
It could have actually been a different manufacturer.   It could have been an H.D. Folsom import marked C. Parker.   Or, it could have been a VH or Trojan grade Parker.   Without the serial number or perhaps an image your mission will end. 
 
Maybe a family member has an image.   Maybe an image is tucked away in a book or bible?   I would start with family members.   Sometimes the least likely family member has an image of that family member that you didn't know of or had never seen.   Good luck.
		 
		
		
		
		
		
		
		
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