The Thwing book does not discuss the parentage of John Livermore, but there is an interesting article that builds a good case that his father was Daniel Livermore and his mother was Elizabeth Hammond.The citation is:
Robert Charles Anderson, New England Historical and Genealogical Register: The English Origin of John Livermore of Watertown and New Haven, Vol. 150, pp. 433-435.
I found this periodical at my local library.
Livermore, California was founded by a Robert Livermore in the mid-1800's, so he would have been several generations removed from the aforementioned people.I have never heard anything of exactly how he was related to the Livermores of America, if at all.