I do not find his birth in the Dedham VR, just his baptism.
Oliver and Abigail were third cousins through James Thorp and Hannah Newcomb.
Sources
Bullard Family Bible and records at Bethel, VT Richard Bullard says born at Sharon, Norfolk Co., MA.
Dedham Vital Records copied by Mary Arris
Ancestry.com, Massachusetts, Town and Vital Records, 1620-1988 (Provo, UT, USA, Ancestry.com Operations, Inc., 2011), Ancestry.com, http://www.Ancestry.com, Record for Oliver Bullard. [1].
Ancestry.com, Vermont, Vital Records, 1720-1908 (Provo, UT, USA, Ancestry.com Operations, Inc., 2013), Ancestry.com, http://www.Ancestry.com, Record for Mark Bullard. [2].
Ancestry.com, Vermont, Vital Records, 1720-1908 (Provo, UT, USA, Ancestry.com Operations, Inc., 2013), Ancestry.com, http://www.Ancestry.com, Record for Oliver Bullard. [3].
Cox, Fred G. The Illustrated Historical Souvenir of Bethel, Vermont. Bethel, VT: 1895, p. 94. Available at ancestry.com.
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DNA Connections
It may be possible to confirm family relationships with Oliver by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Oliver: