You may mean Barnaby River in Nelson Parish, Northumberland Co., NB.
A Patrick Joseph Sullivan b. ca. Barnaby River, Nelson Parish, Northumberland Co., NB 12MAR1914, m. Rogersville, Northumberland Co., NB 27OCT1942 Bertha LeClerc (b. Rogersville ca. 1925), daughter of Joseph and Lina (Gaudet) LeClerc. He was the son of Patrick Joseph and Mary Ellen (Hetherington) Sullivan. Patrick Joseph, Sr. (b. Barnaby River, Nelson Parish, Northumberland Co., NB ca. 1879) and Mary Ellen Hetherington (b. Blackville, Northumberland Co., NB ca. 1881; d/o William and _____ (O'Brien) Hetherington) were married in Northumberland Co. 22JUN1909. He was the s/o Dan'l and Catherine (Casey/Cassey) Sullivan. I think Daniel was the same b. Semiwagan Ridge ca. 1840, and d. Semiwagan Ridge 10JAN1909. Certainly, in 1901, in Nelson Parish, Daniel Sullivan (1JAN1837) and Catherine (4APR1850) had amongst their children Patrick (b. 11JUN1880) and this seems to be the right family and the same Daniel. The problem is that there were several Daniel Sullivan's in the same general area. However when Daniel Sullivan m. 9NOV1875 Catherine CAsey, his residence was in Semiwagan.
I also believe that Daniel (b. 1JAN1837 or ca. 1840) was the same in Nelson Parish in the 1851 census b. ca. 1841, s/o David (ca. 1806) and 'Kathn' ((Catherine Ryan b. ca. 1805) Sullivan, who arrived, probably shortly after being married, from Ireland in 1831. Daniel was baptized 22DEC1839 so he was born between 1JAN1837 and 22DEC1839. the 1901 census is often wrong so I suspect he was b. 1JAN1839. See also: http://freepages.genealogy.rootsweb.ancestry.com/~kvgene/transcrip/crossref4.htmhttp://freepages.genealogy.rootsweb.ancestry.com/~kvgene/transcrip/crossref4.htm
More recent information on the family might be a breach of confidentiality.