The Genealogical Record of the Barnum Family shows him as the son of Ephraim Barnum (1766-1860) and at least two of his children as the children of Elijah Barnum (1786-1864), son of Jehiel.
In the 1850 US Census for Danbury, Fairfield County, Connecticut Elijah Barnum was enumerated with the family of Ora Dikeman, as follows:
Dwelling #526: Family #621
Ora Dikeman, 60, F, b. Connecticut
Francis A. Dikeman, 18, F, b. Connecticut
Julia Dikeman, 10, F, b. Connecticut, Attended school within the year
Elijah Barnum, 70, M, Carpenter, b. Connecticut
Clara Barnum, 68, F, b. Connecticut
James G. Hogan, 23, M, Hatter, b. Connecticut
Charles Begelow, 26, M, Hatter, b. Connecticut