I believe that Beatrice's parents were
James Foster born 1857 Bridge Carston Lincolnshire (cant find where this is) There is a James Foster born 1857 and registered Louth Lincolnshire. Amongst the places covered by the Louth registration district is Castle Carlton, wonder if that is it.
Elizabeth born 1858 Easton Northamptonshire.
Interestingly, in 1901 they were living as boarders in New Street, Claylane, Chesterfield, in a house next door but one to Enoch Baugh, Agnes Baugh and their one year old son, James. So it is possible that the two families knew each other, maybe James Baugh and Beatrice May Foster played together as children.
If you want to follow this family back any further, I think you would need to get Beatrice May's birth certificate to make sure this is her father and to find her mother's maiden name.