Sorry can't help you with the parents. Hopefully someone will see this.
Not my family but this is what I know about him;
MARTIN DUNPHY 1st tried at NEW COURT (Old Bailey) Saturday 1 December 1832. Fourth Middlesex Jury, before Mr. Common Sergeant, 96.
MARTIN DUNPHY was indicted for embezzlement; to which he pleaded GUILTY gave his age as 26. Confined Six Months .
He was tried again at Central Criminal Court(Old Bailey) on the 4 July 1836 and sentenced to Life, (I can't find this trial) transported on the EDEN 1. Master, Alexander L. Mollison Surgeon, Gilbert King. Sailed on 31 August 1836 from Portsmouth carrying 277 male convicts and arrived on 22 December 1836 in Van Diemen's Land.
He married Maria SMITH on the 28 May 1849. Maria died at an aged care institution in New Town, Tasmania on the 10 April 1896 aged 81. she had been born in Ireland
1st child, a son Charles Octavius Cassius Dunphy born 16 November 1849
one of those names might be the grandfather's! I think I'd go with Charles.
Not sure of the order of the children or how many, but there was also a Constantine Dunphy, who died 20 October 1893, at Prince Alfred Hospital in Sydney
Patrick Dunphy ?

His Death Certificate does not give parents; it just states died 22 November 1870 in Harrington-street of throat cancer. age 70. Born in England. Occupation, clerk.
Informant was Joseph McLoughlan the undertaker of Argyle-street.

All the Hobart newspapers carried death notices none referred to family
The Mercury (Hobart, Tas.) Friday 25 November 1870
DEATHS.
DUNPHY —On 22nd November, at his late residence, Harrington-street
Martin Dunphy, aged 70 years.
The funeral will move from St. Mary's Cathedral, this day, at
half past two o'clock, when friends are respectfully invited
to attend. R.I.P. Otago papers please copy.