Town of Rowley records show John Smith as son of Joseph and Sarah.
Birth Records 1737–1835, Marriage Records 1737–1835, Death Records 1737–1835, Declarations of Intention 1737–1835
John Smith son of Joseph & Sarah his wife born December [unknown symbol] Tenth Day. 1751
From StephenAClarke73 on Family Search: Professor John Smith (d. 1809) was said by his wife to have been born in Byfield parish on December 21 in either 1751 or 1752 to Joseph Smith. His mother was a Sawyer. Town records have a John Smith born in Rowley on December 10, 1751 to Joseph Smith and Sarah Sawyer. If you convert the Julian calendar to the Gregorian calendar, the date becomes December 21, 1751, which is a close match. The baptism date, place, and father are all consistent with this being Professor John Smith (d. 1809). This was also the conclusion of John Louis Ewell in The Story of Byfield, a New England Parish (1904), p. 141.