Craft doesn't seem to be around, so I hope you won't mind me posting.
Caroline Street no longer exists but a bit of googling found a reference to it having an intersection with Clement Street. That, in turn, is now a small cul-de-sac off Nunnery Lane, shown here:
http://maps.google.co....001089,0.002411&z=19
Entering the postcode for Clement Street into the 'A Church Near You' website suggests that it's in the Parish of St Clement with St Mary (although, strangely, St Clement's Church doesn't appear to be within actual the parish boundary):
http://www.achurchnea...code=YO231YD&access=1
This site might also be helpful:
http://www.genuki.org...S/ARY/York/index.html
Chris