I remember a similar story of children born in Ireland to English Parents and the reason seemed to be that the father was a soldier who was stationed there.
I checked on Dungavarn and found one quote stating that Dungavarn Castle was used as barracks for the British army from the 18th century.
Some trees on Rootsweb give Mary's parents as Robert and Sarah and that she was born in Dungavarn in 1799. However, Robert married Sarah Pulford in Stamford in 1799 and their daughter Mary was baptised there in 1799.
When Joseph Bunning married in 1813, one of the witnesses was Charlotte Bossingham, the wife of Joseph Bossingham. Joseph's sister, Jane Bossingham married William Bunning in Stamford in 1790.
The families seem to be connected with each other but I can't see exactly how and I can't pin down baptisms for Robert and William. I thought they might have come from Empingham, near Stamford but Robert, b 1778, would be too young to be the father of Joseph and Mary, and is in Empingham in 1851.
Joseph Bunning, who died in 1817, might be the father but I can't find his baptism or marriage.
David