Thanks, Ken, for the rapid & helpful response. Unfortunately there's no genealogical link to Joseph Parker (died 1831) on the MI. His standing is pretty much confirmed by a ref to a Joseph Parker, churchwarden, in the early C19, who looks like the same. Otherwise there is quite a family of Parkers, including a father and son Joseph, in the rest of the century, and all the men living in Dundalk (as far as one can tell) were shoemakers, though the family had enough money to erect further tombstones and it seems likely that some were servants of Earl Roden, the lord of Dundalk. Shoemaker Parkers go back to before 1815 when a Jane Parker 'daughter of shoemaker Parker' was buried at the church, but I know nothing certain of her parentage.
My father's grandfather, John Parker, was born at Dundalk in 1824 (he said, in Dundalk House), the son of Joseph Parker a shoemaker. At some date he joined the Royal Sappers & Miners in which his trade was House Painter, his rank Private. I have not yet seen his service record. This scarcely points to any cavalry tradition in the family!
Is it possible that the 6th Dragoon Guards would have included tradesmen such as bootmakers in their complement? Or perhaps Joseph Parker the cavalryman could have been apprenticed in the boot and shoe trade before enlisting, since he was 21 or so when he joined up? I appreciate this is all piling Pelium on Ossa but obviously I want to make the best use of what info there is! Meanwhile I'll look further into the history of the 6th DGs as you suggest. Anthony Parker