The birth of a Mary Ann Pallister was registered at Gateshead in the December quarter 1838. I don't have any other likely candidates from that period in that area. Large though Gateshead was, it wasn't a hotbed of Pallister activity. Joseph Pallister and Annabell Couper were married there 29 April 1837. A Joseph Pallister's death was registered at Gateshead Sept qr 1840 and an Ann Pallister's marriage was registered there March qr 1844.
The 1901 census (online at FamilySearch.com) gives Jane Palliser of Lamesley as born in Edinburgh, (Sunderland!) about 1856. Her husband does not appear on that census. She also appears in the 1881 census at Eighton Banks, Lamesley, Co Durham with husband and children, Mary Ann age 3 and Alexander age 1, both born Lamesley. Husband John was a stone quarryman age 23 born Heworth. Son Alexander, appears in the 1901 census as McKenzie Pallister, a 21-yr-old coal miner hewer, born Eighton Banks. Also on the 1901 census are Mary Ann age 23, Susan age 14, and Jemima age 4, all born Eighton Banks. A Jane Stewart Forbes was baptised 26 Dec 1855 at Renfrew, Scotland, daughter of Ninian Forbes and Elizabeth Howat Galt (FamilySearch.com), which does not agree with a stated birth of Edinburgh. The GRO indexes show the births of a couple of John Pallisters in the Durham registration district in March 1855 and Dec 1855. Either (or none!) of these could be your man.