According to Mayflower Births and Deaths, she was the daughter of John's first wife, also an Elizabeth.
According to the 2002 Alden Silver book, Mary was the daughter of John and Elizabeth (Phillips) (Everill) Alden.  A problem arises because John and Elizabeth Phillips were not married until 1 April 1660.
↑ Roser, Susan E. "Mayflower Births and Deaths." Vol 1,& 2 Baltimore, Md.: Genealogical Publishing Co., Vol 1 p. 24. Accessed at Ancestry ($).
↑ Esther Littleford Woodsworth-Barnes, Mayflower Families through Five Generations, Vol 16, Part 1, The Descendants of John Alden, General Society of Mayflower Descendants, 2002. "The Silver Book" Page 536
↑ Boston, MA: Inhabitants and Estates of the Town of Boston, 1630-1822 (Thwing Collection). Inhabitants and Estates of the Town of Boston, 1630–1800 and The Crooked and Narrow Streets of Boston, 1630–1822. CD-ROM. Boston, Mass.: New England Historic Genealogical Society, 2001. (Online database. AmericanAncestors.org. New England Historic Genealogical Society, 2014.) https://www.americanancestors.org/DB530/i/14226/14571/260128979
Woodworth-Barnes, Esther Littleford and Williams, Alicia Crane, Mayflower Families through Five Generations, Vol 16 Part 1 of 3, John Alden, Boston, Mass.: General Society of Mayflower Descendants, 2002.