I too have been researching Beadon's parents (Beadon is my gtgtgtgf) and was delighted to find others researching the same people for a change!
I have Beadon and Lavinia's marriage certificate (incase you haven't got a copy yet) - it says Beadon's father is WILLIAM THORN (not Davey). He was a LABOURER. Beadon's mother is not given. Also, Lavinia's father is JOSEPH PALMER (also a LABOURER). They were married 1 July 1849 at the Parish School, Nether Stowey, Somerset. Beadon and Lavinia signed the certificate with a cross.
I also found Beadon in the 1851 census (as Beeton and Leveimier!). This census said he was born at Lilstock, Somerset. So I googled Lilstock and found a pile of transcribed Lilstock BMDs. Beadon was baptised at Lilstock on 18/12/1825. On the baptism record it said his parents were MARY DAVEY (no father given). Earlier in the record (1819-1823) there are four children born to EDAWARD/MARY DAVEY. Two of these children are CHARLES DAVEY (bapt 1820) and ISAAC (bapt 1823) - are these the two Beadon was living with in 1841 in Nether Stowey? Maybe.
I had a look at the marriage records and found that an Edward Davey married a Mary Coles on 19/11/1818 at Lilstock. So that explains where the name Mary Coles comes from, but it still gives us no direct link to Beadon. I wondered whether Edward Davey died before Beadon was born (and hence why he is not given on the baptism record), but the burials record for Lilstock stop at 1812. So haven't yet found this info.
Curiously, there is also an ANABELLA DAVY (no e) baptised at Lilstock in 1828 and again MARY is only listed as the parent. Is this the same Mary as Beadon's mother? Don't know. On the baptism record it appears there is another Davey family spelled DAVY - so Beadon and Anabella may not be related.
Hope this helps. To google the Lilstock records type in Lilstock registers. It should come up with a
www.wsom.org.uk website. Find Lilstock (and Nether Stowey, Over Stowey) in there.
I would be keen to hear if you find more info about the mystery of Beadon's father. I cannot find a William Thorn anywhere.
Cheers, Fiona.