A possible, which also might explain why details of his parents' are sparse in Australia:
I can find a reference at
www.nationalarchives.gov.uk/a2a to a William Pitts, artist, who in 1830/1831 was at 5 Watkins Terrace Ebury Street Pimlico.
In 1841 there are the following people at Watkins Terrace (note, no relationships given, ages 20 and up rounded down to nearest 5 years)
HO107/0732 9/5/3
Mary Pitts, 45, occupation "independent" born out of county
Thomas, 25, artist, b. in county
William, 20, sculptor, b. out of county
Mary, 20, b. in county
Joseph, 20, sculptor, b. in county
Frederick, 9, b. in county
William Pitts, "the sculptor" of 5 Watkin's Terrace is recorded as having committed suicide in April 1840 (if you have access to the Gale database, you can find these records). He took laudanum, and from the reports it sounds as if he was suffering from depression (verdict was "temporary insanity"). William was said to be around 60 years of age, and Thomas Pitts is mentioned as his eldest son.