Proving Parentage of Alexander Neeley Bierce

Hello,

My cousins and I are missing one very important piece of documentation in order to be eligible for the Mayflower Society. We need to prove that Alexander Neeley Bierce (Bearse, Bearce, Bears, Beers, Pierce, etc) b. 1798 in Massachusetts and d. 1853 in Illinois is the son of Austin Bierce, Jr. b. 1769 Cornwall, Litchfield, CT, d. 1839 Claverack, Columbia, NY and Zerviah Bryant b. 1770, d. 1839 Claverack, Columbia, NY. This family seems to have started in Cornwall, Litchfield, Ct, migrated to Sheffield, Berkshire, MA and then settled in Columbia Co., NY. They may also have lived in Greene, Dutchess or Washington Co.s. Any leads most appreciated.
Eleanor