This is from familysearch wiki:
Parish Records
Durham University Library Archives and Special Collections Reference number: DDR/EA/PBT/2/227 Date: 1763-1891 Contents: Including transcripts from South Shields Holy Trinity, September 1834-1848, and South Shields St Stephen, October 1846-1848 DDR/EA/PBT/2/226 Parish Register transcripts are available to search free online at FamilySearch Historical Records.
The dates of the post-1760 transcripts have been noted in detail and sometimes only cover years. For most parishes in the collection there are gaps in the sequence of transcripts. It is advisable to consult the original parish registers for these years and events.
The Parish Registers for the period 1653-1953 are deposited at Durham County Record Office, County Hall, Durham, DH1 5UL (EP/SS.SH).
So it sounds as if the baptisms for the years you're looking for are not on familysearch.
I haven't checked the years findmypast hold, but I suspect the same thing.
So Parish regs are at Durham County Record Office.
It's also possible they were non-comformists.
There are three places of worship belonging to the Wesleyans; two each to the Baptists, Presbyterians, and Methodists of the New Connexion; and one each to the Independents, Primitive Methodists, and members of the United Secession Church.