I assume you already have Joseph, a shipsmith, and Margaret in 1841, living at Stockport St. in Middlesbrough, and you already have Margaret living with widowed daughter Catherine Walton and assorted grandchildren (two of daughter Catherines children and two of son Johns children) in Middlesbrough in 1861.
Joseph and Margaret are living in the same street in 1851 as in 1841:
1851 census. HO107/2383 p.40.
Stockport St., Middlesbrough,
Joseph Pearson, Head, 67, Shipsmith, b. Northumberland, Newburn
Margaret Pearson, Wife, 70, b. Northumberland, Hartley
John Walton, Visitor, 7, b. Middlesbrough.
I know Margaret is down as being born in Shields, Durham in 1861, but the census records often have mistakes, she obviously lived in Shields a long time as her children were all born there.
From the South Shields St. Hilda parish records :
Margaret Pearson, born Nov 28th 1805, baptised June 25th 1806, 2nd daughter of Joseph Pearson, Blacksmith, native of Newburn, Northumberland, by his wife Margaret Mustard, native of Hartley, Northumberland.
Looks to be one of several children of Joseph Pearson and Margaret Mustard baptised in South Shields. This looks to be your Joseph and Margaret and family.
Alan.