Is that the same man as James? Here are some sources:
"United States Census, 1860", database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MFDZ-XXD : 12 December 2017), Flavel Parker, 1860.
"United States Census, 1880," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M8WJ-CBX : 15 July 2017), Flavil Parker, Guilford, Windham, Vermont, United States; citing enumeration district ED 228, sheet 377A, NARA microfilm publication T9 (Washington D.C.: National Archives and Records Administration, n.d.), roll 1349; FHL microfilm 1,255,349.
"Vermont Vital Records, 1760-1954," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:XFK4-SVQ : 18 October 2017), Flavel Parker and Alma Field, 03 Dec 1824, Marriage; State Capitol Building, Montpelier; FHL microfilm 27,648.
"Vermont Vital Records, 1760-1954," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:XFK4-7X3 : 18 October 2017), Flavil Parker in entry for Joshua Flavil Parker, 27 Sep 1832, Birth; State Capitol Building, Montpelier; FHL microfilm 27,648.
It looks like Flavel and Alma had more children than just Joshua and Alpheus, too.