There is a Mark Parker, tobacconist, shown in various historical directories in business at 30 Southgate Street, Bath, between at least 1923 and 1935 which feels like the right era for this picture, judging from the gentleman's clothes.
No M. Parkers shown as hairdressers around this time anywhere in Somerset as far as I could find. Could this be it?
There's a very small picture of the present 30 Southgate here which looks like it might be.
Actually it seems to be on the non-pedestrianised bit of Southgate (marked on Google maps as St James' Parade) so you can see it in Google Streetview: (hope this link works)
Otherwise go to Google Streetview, find St James' Parade in Bath, find the Post Office and its 2 doors to the left, next to Toni & Guy....
There's little enough detail to be sure but the general layout looks similar, and the corbel things are visible in both pictures, as are what I guess are iron pillars in the angle of each window....
That's brilliant! - thank you so much - the person who sent the photo will be delighted. I'm sure you are right and very observant! They thought that it was in Weston so I didn't look anywhere else. Thanks for the links - they worked very well and showed enough detail to confirm your detective work.