All of this info was found in the Chelmsford, Middlesex, Massachusetts vital records:
Birth...Proctor, Hannah d. Peter and Hannah Jan. 10, 1722-3 (that's exactly as it reads and through all of Chelmsford I find no record of any Hannah with parents of James and Judith under any of the spellings of Proctor, so if we are sure of her birth date, and birth place, I'm pretty darn sure this is our Hannah)
Marriage...Procter, Hannah, and Moses Barrat [of Nottingham int.] (that just means their intention of marriage was recorded and that he was from Nottingham, they often filed the intention in both locations) Nov. 11, 1742
Marriage...Barrat, Moses [of Nottingham. int.] and Hannah Procter, Nov. 11, 1742
Birth...Barit, Moses, s. Thomas and Rachell, Feb. 1, 1718-19.
Marriage...Procter, Peter, and Hannah Harwood, int. Apr. 10, 1720.
Death...Prockter, Hannah, wid [Peter.P.R.4] bur. Oct. 2, 1781, a. 81 y. C.R.
1. [d. Sept. 30. P.R.4]
Death...Proctor, Peter, Feb. 20, 1772 [a. 78 y C.R.1]
I just looked up James Proctor and Judith Nichols to see why they were listed as her parents on some websites. They are from Woburn , Mass, and I found vital records showing the birth of their daughter, Hannah Proctor b. 2 May 1727 in Woburn, Middlesex, however, I also found vital records showing she married a Joshua Reed 9 Jul 1742 in Woburn, and no record of marriage to a Barrett. So I think we can conclude that our Hannah's parents were Peter Proctor and Hannah Harwood.