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That hallmark date letter is for July 1938 to June 1939. That tallies well with the serial which also dates to ~'38.
Being as this was cased up in Birmingham, it wont necessarily have a conventional case number unless there was an identical Swiss cased equivalent you can find.
Yes by case number I mean reference number. The number on the one you show wont be an Omega case number or serial, it is internal to Dennison (who have been defunct for decades). On post war models there is usually a Dennison model number and a serial number stamped in the caseback. I am not certain which that is, probably just a serial number which will be useless for identification. Usually Dennison stamp the last 3 digits of the serial (not model) onto the movement spacer or case. You can just see it on the pics above.
You have probably researched this but in case not then the Parsons on the dial would refer to Parsons jewellers in Bristol which has been around for hundreds of years. It is still open today i think although it changed name a few years back to foxhills.