Richard MASON is likely to be the father to Doctor Richard MASON. There is a reference to Bury St Edmunds in the "History of Hinckley" in a section about the Doctor, and there is a baptism record that appears to fit the available information.
A Richard MASON also baptized Alice MASSON in Bury St Edmunds in 1603, and may have moved further south to Finchingfield, Essex shortly after his son Richard's birth in 1615 where a Richard and Grace MASON baptized William (Jul 1618), William (Apr 1619), Anne (1620) and Grace (1623). Finchingfield would be closer to London where his Richard was sent to school. Grace (1623) married Francis BENSON son of Francis BENSON in Finchingfield in 1653.
