Hi Wendy,
Further to Kellie's find, at the Ballarat Cemetery, you could order and pay online, and receive instantly the death certificate(s). They should list the names of their children.
Name: Mary Harrington
Death Place: Ballarat, Victoria
Age: 81
Father's Name: Unknon
Registration Year: 1899
Registration Number: 4405
Estimated Birth Year: abt 1818
Name: James Harrington
Death Place: Ballarat, Victoria
Age: 63
Father's Name: Jas
Mother's name: Kate Walsh
Registration Year: 1884
Registration Number: 165
Estimated Birth Year: abt 1821
Mary