John Belden, b. 1631, Kippax, yorkshire, England & d. June 27, 1677, Wethersfield, Hartford, CT
Lydia Standish b. 1637 & d. 1679 in Wethersford, Hartford, CT
One way to "prove" that Lydia was their daughter is to find their probate papers. Probate courts in Connecticut are on a district level and not a county level. You can contact the Office of Probate Court Administration, 860-231-2442 and ask which specific court to contact.
According to one family Tree, Stephen's parents were Joseph Kellogg b. March 1626, England and died 1709 Hadley, Massachusetts.
His second wife Abiail Martha Terry was Stephen's mother. Abigail was born 1646 in Windsor County, CT and died before 31 Oct 1726, Hadley, Hampton, MA.
When I was looking up my East Coast ancestors, I found American Ancestors to be a wonderful help. Back then it was New England Historical & Genealogical Society and was a pay site but worth the money. Successful Searching!