Based on the 1851 census it appears that William's father John was himself the son of a farmer in Kepwick...
KIDSON, John Head Married M 49 1802 Farmer Kepwick, Yorkshire
KIDSON, Margret Wife Married F 30 1821 Melmerby, Yorkshire
KIDSON, William Son M 12 1839 Scholar Kepwick, Yorkshire
KIDSON, Mary Ann Daughter F 11 1840 Scholar Kepwick, Yorkshire
KIDSON, John Son M 2 1849 Kepwick, Yorkshire
COULSON, Mary Servant Unmarried F 19 1832 General Servant N K, N K
BEADLOW, William Servant Unmarried M 30 1821 Plough Man Redmile, Yorkshire
HUTCHINSON, John Servant Unmarried M 15 1836 Plough Boy Borrowby, Yorkshire
Piece: 2371 Folio: 472 Page: 10 Registration District: Thirsk Civil Parish: Kepwick
Address: Furlands House, Kepwick County: Yorkshire (North riding)
It's harder to be certain about William's mother Elizabeth's origins
(you will need to confirm her father's name from her marriage certificate).
She was probably either born in Windermere as daughter of Luke Sowerby (basketmaker)...
SOWERBY, Luke Head Married M 41 1810 Basket Maker (Master) Employing 2 Man & 1 App Colton, Lancashire
SOWERBY, Margret Wife Married F 38 1813 Colton, Lancashire
SOWERBY, Elizabeth Daughter F 11 1840 Scholar Windermere, Westmorland
SOWERBY, Agnes Daughter F 9 1842 Scholar Windermere, Westmorland
SOWERBY, John Son M 7 1844 Windermere, Westmorland
SOWERBY, Mary Daughter F 5 1846 Windermere, Westmorland
SOWERBY, Luke Son M 1 1850 Windermere, Westmorland
HARDEY, Robert Unmarried M 22 1829 Basket Maker Heversham, Westmorland
TAYLOR, Thomas Unmarried M 22 1829 Basket Maker Not Known, Westmorland
AIREY, William Aprentice M 15 1836 Basket Maker App Windermere, Westmorland
Piece: 2441 Folio: 179 Page: 22 Registration District: Kendal
Address: High Cleabarrow County: Westmorland
or ALTERNATIVELY in Warcop as duaghter of a farm labourer...
SOWERBY, William Head Married M 34 1817 Agr Lab Appleby, Westmorland
SOWERBY, Elizabeth Wife Married F 27 1824 Smardale, Westmorland
SOWERBY, George Son M 5 1846 Bleatarn, Westmorland
SOWERBY, Elizabeth Daughter F 4 1847 Warcop, Westmorland
SOWERBY, Anthony Son M 2 1849 Warcop, Westmorland
Piece: 2439 Folio: 202 Page: 7 Registration District: East Ward
Address: Warcop County: Westmorland