That would be the Reading Parker line (which I believe is Family Group 19). I call them that to differentiate them from the Barnstable Parkers, the Middlesex County Parkers (Family Group 10), and the ones in Andover (who I have not tried to figure out yet).
The Reading Parkers descend from Deacon Thomas Parker that you identify. The Middlesex County Parkers from the 5 Parker brothers (James, Abraham, Joseph, Jacob, John, the last childless). I haven't really dug into the Barnstable or Andover Parkers, but I run across them periodically.
There is a Parker line in Saybrook Ct, one on Parker Island in Maine, one in Nansemond(sp?) Va., another group in New Jersey, I bump into them occasionally and many others.