Hi there.
Well I've found George in the 1861 census on ScotlandsPeople ( or George <i><b>Dimbar</b></i> as they have him indexed as!).
No sign of his parents though. He's listed as a boarder in the house of a Mary McRobert. However the good thing is that Mary's mother is also in the house, and she is Esther Gordon aged 50. I assume then that Esther is George's grandmother.
No sign of George's father, James, though, at least not in Kirkcudbright.
I'm speculating the George's mother, Elizabeth Gordon died when he was young, possibly even in childbirth as I haven't been able to find a death cert. I know that she was deceased by the time of his marriage in 1876 and I have not been able to find her death on ScotlandsPeople.
As to his father still no further along. I believe that it is the James that I found on the 1851 census, but even if it is that puts up another stumbling block as James' father Joseph, is listed as being born in Ireland.
So unless I can find more information or have a sudden burst of inspiration, then I guess that's as far back as I can go with the Dunbars.