When Una Ann Cadby married her father is given as Joseph Cadby a toolmaker.
Cadby is a relatively unusual name.
Prior to this in 1871 Una A Cadby is living with her parents Joseph and Mary Cadby at Park Road, Edward Place, All Saints, Birmingham.
This seems to make it almost certain that Una Ann's parents were Joseph and Mary.
It has not been possible to find the family in the 1861 census but the 1851 census backs this up.
Una Ann's birth certificate confirms her parents as Joseph Cadby and Mary Cadby, formerly Curtiss.
Thus the link to Joseph and Mary as her parents is fairly certain and that Mary was surnamed Curtis.
They appear in the 1841 census and their marriage has been identified.
Una Ann was also the informant for the death of her mother in 1871, 2 years before she married.

When married Mary's father is given as John Curtis.

The 1851 Census gives her birthplace as Ashwell, Herefordshire but later research revealed that she was born at Ashwell, Hertfordshire.
Her birth was not registered until June 1837 at the Calvinistic Baptist church in Wolston, Warwickshire. Her name was given as Mary Barbara Curtis. Her sister Unna Ann Curtis was registered at the same time.  Her parents are given as John Curtis and Sarah Curtis (formerly Oliver).
The full name of Mary Barbara Cadby was also used on her death certificate.

Mary's identity therefore seems certain.