I would take a look at that 1861 (RG9/3848 folio 118 page 13) census again--It looks like Joseph Mason and family were born Shield Row--that is Tanfield. The the 1851 (HO107/2391 folio 360 page 33) says that the children were born in Halgarth which is in the parish of Pittington.
Confusing. I looked in parish records for Pittington, Penshaw, and Tanfield. No baptisms of these children. Perhaps they were not C of E.
The 1851 has a daughter Jane born abt 1832 Chester [le Street] but no baptisms there either.
No early marriages in Co. Durham or Northumberland of a Joseph Mason.
marrying a Margaret. Yes I would get a birth cert if you can.
Janis