Her birth record has not been located (and is not listed along with the records of Lydia and Jonathan her siblings). However, her father is known to be Henry Taylor of Barnstable, information that comes from the Will of widow Mary Haughton, wherein it says: "To Mary Taylor ye Daughter of Henry Taylor twenty Shillins". The will is found printed in the Mayflower Descendant 18:134-5 (1916), and is dated Jan 19, 1686.

"Cape Cod Series, Vol II, History and Genealogy of the Mayflower Planters and first Comers to Ye Olde Colonie" by Leon Clark Hills; Hills Pub. Co., Washington D.C.; 1941, pg. 83, also lists Mary as the daughter of Henry Taylor of Barnstable. So does [Freeman 2:

better source. The book is online at archive.org
"Descendants of Edward Small of New England" by Lora Altine Woodbury Underhill
Volume III
Footnote on Page 1198

Henry Taylor mar. Dec. 19, 1650, Lydia Hatch ; he mar., second, Mary
ISSUE BY FIRST WIFE
I. Lydia Taylor, b. June 21, 1655, probably in Barnstable.
II. Jonathan Taylor, b. April 20, 1658, probably in Barnstable.
III. Mary Taylor, b. about 1660, probably in Barnstable; she was mar. Dec, 1686, to Doctor William Dyer, of Truro.
ISSUE BY SECOND WIFE
IV. Hannah Taylor, b. July 7, 1665, in Boston.
V. John Taylor, b. Aug. 4, 1666, in Boston.
VI. Mary Taylor, b. June 6, 1668, in Boston. It was not unusual to have two children of the same name, especially if they had not the same mother. (Vide Samuel Dyer, son to Dr. William Dyer, who had
a Mary by his first wife, and another Mary, or Molly, by his second
wife ; both reached maturity.)
VII. Henry Taylor, b. Oct. 12, 1670, in Boston.
Vide New England Historical and Genealogical Register, vol. 2 : 66 ; also Commissioners' Records of Boston, vol. 9 : 97, l01, 109, 116.