I believe that Hananiah, son of Deacon Thomas, never lived in Chelmsford, MA. At least I don't have source for it if he did. I also do not have a Josiah as a child of Hananiah and Elizabeth Brown(e) in any of my sources or Parker family books.
I'm wondering if Josiah is a son of one of the Parker brothers from Great Burstead, England. There are so many similar names among these brothers and the sons of Deacon Thomas.