Hi Genie - Unfortunately Tasmania does not seem to be in line with most of the other States when it comes to BMD Indexes.
There is an online index available for BDMs up until 1899 and a search for Richard shows exactly the information you have and no more.
I don't think the marriage certificate would be much help to you either.
I have a Marriage Certificate for a marriage in Tasmania in 1884 and there is no provision on it for the recording of parents names.
A Full Death Certificate would give that information - IF it were known by the Informant but that is a gamble.
However,
presuming Richard Henry BEARE was born ini Cornwall then try this:-
Gulval PR Transcripts - Baptisms
1863
25 Feb William James c/o James (Labourer) & Anne Beare of Punjoe private baptism
2 May Christiana c/o James (Labourer) & Anne Beare of Punjoe
2 May Sarah Ann c/o James (Labourer) & Anne Beare of Punjoe
1864
18 Oct Richard Henry c/o James (Labourer) & Ann Beare of Punjoe private baptism
1867
18 Dec Marianne c/o James (Labourer) & Anne Beare of Penzance
1871 Census - Marazion
Lay's Lane
James BEAR, head, mar., 37, agricultural labourer, Ludgvan
Hannah do., wife, mar., 34, Camborne
Amelia J do., daur., 14, scholar, Ludgvan
Christiana do., daur., 11, scholar, Ludgvan
Sarah A F do., daur., 10, scholar, Ludgvan
William J do., son, 8, scholar, Ludgvan
Richard H do., son, 6, scholar, Ludgvan
Mary H do., daur., 4, scholar, Ludgvan
George do., son, 1, Marazion
This may be the marriage from FreeBMD:-
Surname First name(s) District Vol Page
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Marriages Dec 1857 (>99%)
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Bear James Penzance 5c 532
Carter Hannah Penzance 5c 532
Hosking Mary Penzance 5c 532
Penberthy James Penzance 5c 532
I think that should answer much of your question!
Strange - I cannot find a record of Richard's birth in FreeBMD!
(I have seen that problem before though with a Christopher Trewhella for whom I also found a baptism.)
I'm also not sure where the marriage took place but I cannot find it (at least so far).
It certainly did not happen at Gulval and I do not have records for Ludgvan and it was also not at Camborne.
Anyway - hope this helps. It certainly shows a Richard Henry as a brother to a George BEARE so that's has to be a positive.
CT