"Massage parlour" has normally been a euphemism for a brothel, where in most cases you were highly unlikely to get a massage! If we are talking about Thai and Chinese places with a shop front and a legit clientele (at least for the Thais), I prefer, along with some others, to stick to the term massage shops, especially as the girls themselves refer to them as shops, for places where you can get an actual massage, probably with extras (almost certainly with the Chinese).
Sticking to this basis, there seemed to be a sudden increase in Thai shops in Cardiff about 2015. I have no idea why. The Chinese in this form seen to have arrived later here, although you could find Chinese FS providers who could also do decent massage before that. I am only familiar with South Wales, mainly Cardiff, to some extent Swansea and Newport in this regard, so they may have spread earlier elsewhere. There is a Chinese place in Newport that seems to have links with Birmingham.