Many thanks to all who have responded.
The book " An Emigrant's Home Letters" is well worth downloading and reading for its accounts of life in poverty in London in the late 1830's and Sydney in the 1840's, and shipboard life during the 4 month voyage. It's one of those books that is hard to put down.
This is what I've discovered so far:
Henry's Parents
Father: Thomas Parkes, born Stoneleigh, Warwickshire c. 1780 (probably to Edward and Sarah Parkes in 1778). Tenant farmer, migrated to Wales and then back to Birmingham, spent some time in debtors' prison. Died c. 1860
Mother: Martha Faulconbridge born Allesley, Warwickshire 1774 to Thomas Faulconbridge and Mary. Died 1842 after a long illness.
Thomas and Martha were married 12 October 1800 in Stoneleigh.
Martha may have had children from previous relationships: Allesley parish records in the Coventry Archives contain papers concerning a Martha Faulconbridge “who is with child by Samuel Hadley” (PA 295/55/6; 25 Sept 1790) and a Simon Hyott “respecting Martha Faulkenbridge” ( PA 295/17/6; 31 March 1797). The IGI has the baptism in 1799 of a child Martha born to Martha Faulconbridge.
Census: In 1851, Thomas is with daughters Maria and Sarah, and grandson Thomas Anthony Parkes at 233 Bristol Street, Birmingham. I have not been able to find Thomas or Martha in the 1841 census.
Henry’s Siblings
The ones named in the book are:
Sarah: bap 1801 Stoneleigh. A staymaker, in the census with sister Maria at 233 Bristol Street Birmingham in 1841, 1851 and 1861. Father Thomas is with them in 1851. Nephew Thomas Anthony is with them in 1841 and 1851. Sarah died in 1861 when Henry was in England on a lecture tour.
James: bap 1805 St John Coventry. Had a son William EDIT James was a police constable. Sister Eliza is with him on the 1851 census - back of 88 Heneage Street, Aston
George: bap 1807 St John Coventry
Eliza: bap 1809 St John Coventry; staymaker.
Maria: bap 1811 St John Coventry, died Springwood NSW 1891. Presumably migrated to Australia after death of sister Sarah in 1861
John: nothing known
There would appear to have been one more sibling: the father of Henry’s nephew Thomas Anthony Parkes who was born in Pontypool, Monmouth circa 1829. We know from the book that Tom was an orphan and was the son of Henry’s eldest brother. He was being brought up by Sarah and Maria. Tom’s father was probably Thomas Parkes (bap 1803 St John Coventry).
The nephew Thomas Anthony Parkes married Annie Moore in Sydney in 1855.