Many of my Cole ancestors are in the Salem United cemetery located outside Simcoe Ontario. My 4thG-grandmorther married John Dudbridge, her name is Elizabeth Beemer. I am not sure of the connection with the Barker John and Elizabeth moved to this area just after the 1812 war and had several children, I am still researching this family to see where the Beemers and Dud(d)bridges departed from in the USA - I believe from NJ or PA. I am guessing at the moment that my Elizabeth Beemer could be related to your Barker family somehow.
I would like to see if there is a connection with the two families.