You are right, Parker Hale did not manufacture No4s, but they did sell quite a number over the years. The stamp "PARKER HALE-BIRMINGHAM" indicates that the rifle was first retailed by them, probably sometime in the early 1950s. It was either purchased by an individual and imported, or by one of the Canadian importers like Fraser of Toronto or Nelson Coleville of Winnipeg and subsiquently sold to an individual. Then, in 1963 it was submitted to the DCRA's 7.62 conversion programme. There is one other English retailer stamp that may be encountered as well, "ALF.J. PARKER-BATH ST.-BIRM." The "REGULATED BY FULTONS" marking indicates that the rifle was at one time in the shop of Fultons of Bisley for repair or adjustment work. This could have been either before or after the DCRA conversion to 7.62, but has nothing to do with the DCRA conversion process itself. See my last photo above.
One more thing. Any 7.62mm accessories that are attached to your rifle have nothing what so ever to do with the DCRA conversion. When rifles were submitted for this process, all that was required was the barreled action, and that's all you got back; action,new 7.62 barrel attached plus old .303 barrel returned. Stocking up, sights, etc. responsibility of the owner - whoever that may have been; individual, gun club, Military Rifle Association, Provincial RA or the DCRA itself. The addition of 7.62 magazines, extractors and charger guides didn't make them shoot any better.