I worked my way through all the A Parmenters on Ancestry or FindMyPast in any military record from 1898 onwards. That led nowhere, so out of frustration I clicked on a poorly indexed page on FindMyPast, with just the surname, a DOB of 1871 and Place, Halstead. It solved the puzzle (or at least some of it)! As soon as you view the record, you can see the first name of the soldier and his regimental number in the RA (so no idea why it is not indexed). It has all the details of the service, the DCM, gunshot wound to his side and pension of 18d per day. Sadly, it wasn't Arthur. It was Albert, and I can't even claim a distant relationship to this hero.
Interestingly, Albert (who had not been in the Essex militia) had previously served in the Submarine Miners Militia at Harwich which I had not heard of before.
It doesn't explain Arthur, though. There was one man in the 1st Essex who received a South Africa medal, with clasps for Cape, Orange and Transvaal, and I can't find anything about him (service no 5440). Would Arthur have transferred back to the Essex from the RA for the remainder of the Boer war, after attending the Queen's funeral? If so, why the 1st? If that is not him, and Albert clearly is not Arthur, I can only assume Arthur did not go abroad in the Boer War. The 1901 census says he was in the 92nd battery. It would be interesting to know where 92nd Battery were in 1900-1903.
I am beginning to think Arthur remained on the reserve and was recalled in 1914, but did not go abroad. Some of the Essex were on Home Defence or maybe his job was protected - the foundry made those big cast iron Tortoise stoves and they were in demand for public buildings, including (I would think) hospitals and military establishments all through the war.
Well, that's a weekend I won't get back but I learnt a few things about the Boer War and now have a history of Albert Parmenter if anyone is interested!
Thank you Ken