John, Joseph, Catherine and William were all baptised in South Shields after 1812. Records after 1812 only show baptism date, mothers first name, fathers full name and occupation, and where they lived. So only father Josephs full name and occupation - Blacksmith, and mother Margarets first name are recorded.
Baptism records from 1798 to 1812 were far more detailed, showing birth date, number of child (eg 2nd daughter) mothers name and maiden name, fathers name and occupation, where they lived, and mothers and fathers native parish (although this was patchy).
The 1851 census records that Joseph Pearson was from Newburn, and his wife Margaret was from Hartley. Assuming that this is correct, then it tallies in with the baptism records for South Shields, from 1804 to 1812 for the earlier children of Joseph Pearson, recorded as native of Newburn, and Margaret Mustard, recorded as native of Hartley, on at least 2 of the baptisms I checked.
Alan.