They are on the 1861 census living at Iwerne, Shaftesbury, Dorset
Thomas E Cole born 1827, farmer of 50 acres employing 2 people, born Kingston, Dorset
Anna L Cole born 1834 in Kingston, Dorset
Joseph B R Cole, son, born 1861 in Gillingham, Dorset
This is his birth registration:-
COLE, OWEN JOSEPH ROBERTS (MMN) ROBERTS
GRO Reference: 1860 D Quarter in SHAFTESBURY Volume 05A Page 208
I'm just wondering if the B should have been an O.
The usual place to look regarding trades is Kelly's Directories. I always seem to get into a muddle when trying to search these online, so perhaps someone with more expertise can have a look in a Dorset one for you.