Just digging this up to say hello.
If you're still getting notices of replies ...
My grx2 grandfather and that Samuel who was 15 in 1841 and went to Australia - Samuel Sandoe Bice - were once business partners.
http://www.nationalarchives.gov.uk/a2a/records.aspx?cat=021-cn_2&cid=1-7-1-3-5-2#1-7-1-3-5-2 CN/1997 1859
Draft licence for 21 years to work minerals.
Thos. Tristram Spry Carlyon of Tregrehan, esq., to Francis Hoare Hill
of Finsbury, Mddx., and Sam. Sandoe Bice of St. Blazey, mine agent.
Menagwins and Tregorrick.
I have no idea whether anything actually came of that. Francis is nowhere to be found in 1861 (his family is in Plymouth, and I actually wonder whether he was in Australia, since he seems to have been connected also with a copper-trading family in Wales with history in Australia), and shortly before the 1871 census he went bankrupt in London, as a share dealer.
I would imagine that Samuel's son who became Sir John George Bice in Australia was my gr-grfather's childhood friend, as they are from the same place in Cornwall and were almost the same age. My gr-grfather spent a few years in NSW around 1891 (having travelled originally to Victoria), and then returned to England, and I've never known whether the Bices' earlier emigration there might have been connected to his decision to go there somehow.
Meanwhile, how is it that you have William Bice's exact birthdate and place of birth in 1766 but you seem to be looking for his parents' names? (Edit - I think there's a typo in your information: it is meant to be his death date and place, so it isn't 1766 as he married in 1788.)
There were multiple Bice baptisms in Gwennap 1753-1771, children of Hercules and Jane. They included Williams baptised in 1761 and 1762.
A couple named William and Philippa Skinner had children baptised in Gwennap in 1772 and 1776. I wonder whether they would be worth investigating.
Unfortunately, there may not be conclusive answers, without parents' names on marriage records to connect them with births of the bride and groom.
http://www.cornwall-opc-database.org/If you're looking in!