The mysterious Murphy's of Cork: link to Australia in the goldrush era

I am trying to trace my maternal grandfather's line which is traced back to Cork.

Mary Murphy died in Timor, Victoria in 1888 aged 68 years of age. Whilst in Australia, she married very young (aged 16) to James Walsh around 1836 in Bathurst, NSW. Her death certificate recorded some details, born in Cork, Ireland, and her parents' names were Jeremiah Murphy (unknown profession) and her mother's surname was O'Connor. Since her grandchildren did not know Jeremiah's profession, Mary's mother's first name, I wonder if she came to Australia on her own or if her mother died young. 

I searched Irish baptism databases and found Mary Murphy born on the 5th of May 1820, her father's name was Jerh Murphy and her mother's name was Joha Connor. The civil parish was Kimurry, Co. Cork, and no further information about townlands. These names are the only ones I could find with matching names for Mary's approximate birth year in the whole of Ireland. 

Jerh and Joha had two other children, Julia born in 1813 and Cath Murphy born 1815. I cannot find any further information about the family after 1820. 

Mary had quite a tragic life in the Victorian goldfields, probably quite standard in those days. It would be great to flesh out her life a bit more and I am sorry I do not have more to go on.

I have checked https://www.irishgenealogy.ie/en/ quite a bit but I do not subscribe to any databases like Ancestry. Griffith's Valuation records show a few Jeremiah Murphys in the Kilmurry parish between 1848 and 1852. 

If anyone has any suggestions about databases or recommend further searches, that would be great.

Thanks!