With this profile we find one of those rare Howling Errors from the History of Parliament Online. It wasn't he, but his grandson of the same name, who married Lady Mary Chester, daughter of Sir Anthony Chester, Bart.
This error may stem from the History of Norfolk, whose Bell pedigree is described by the author of "Genealogical memoirs of the extinct family of Chester of Chicheley" as "deficient and inaccurate". (Op. cit., p. 120) With three consecutive Sir Robert Bells, and two of them (the first and third) in Parliament, it's easy to see how the error could have occurred.
There is no record of any earlier Anthony Chester with a daughter Mary.
Sir Anthony Chester had issue by his first wife Elizabeth Boteler five sons and seven daughters, all of whom were born and baptized at Chicheley.
I. Mary Chester, baptized at Chicheley 23d Aug. 1590, married Sir Robert Bell Kt. of Beaupre Hall in Norfolk, who was the cousin-german of Elizabeth Peyton, the wife of Sir Anthony Chester the second Baronet. Sir Robert was knighted by James L on 26th Nov, 1613, (13) and was M.P. for co. Norfolk in 1626 in the second Parliament of Charles I.