Daniel,
Just a wee snippet, to help, or hinder your quest !
As Sharmor says, Barrington lay within the Parish of Bedlington.
Although Bedlington(shire) lay within the boundaries of the county of Northumberland, it was owned/administered by County Durham until 1844.
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If John and Ann were true to the English Naming System, then clues to her parents names are to be found in the names of their own children. ( e.g. second son of a couple was to be named after maternal grandfather.. )
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Michael Dixon