I've tried tweaking the photo a bit to remove some of the colour cast (see below), but I'm still not sure what colour the bus is. Dark blue? Dark green? And I think there's a paler stripe below the side windows.
Anyway, I've been trying to see if it matches any Yorkshire bus operators, but haven't had a lot of success. The following are ones I've found with mainly dark blue or dark green buses, usually with a paler or contrasting stripe or other feature:
East Yorkshire Motor Services (though I note Carol's comment at #6 above)
Rotherham Corporation
Middlesbrough Corporation
Blue Line (Armthorpe, near Doncaster)
South Yorkshire Motors (Pontefract area) - a bit paler than this?
United Services (same kind of area, and also a bit paler?) (Not the United buses from North Yorkshire and further north)
The above were all in various shades of blue; I've omitted Bradford because they were more a mid blue.
Leeds City Transport buses were dark green with a paler stripe, but if the pictures I've found are representative, they didn't have many single-deckers.
Trish - do any of those places/areas seem possible for where your father might have taken photos? And as Ray asked, might he have gone somewhere specially to watch the cycling, or would he have been more likely just to take photos of a big event in his home town?
Arthur
Edit:
1. I also tried distorting the image to see if I could make out the shop name next to Dewhursts, but without much success.
2. The street lights look quite unusual - anyone recognise them?