New South Wales has a great BMD Online...
REG NO. LAST NAME GIVEN NAMES FATHER'S GIVEN MOTHER'S GIVEN
V18434487 47/1843 COULTER ROBERT J DAVID MARY
It would appear that Robert's parents were married in NSW...
REG NO. GROOM BRIDE DISTRICT
V1841922 75/1841 COULTER, DAVID MCNEICE, MARY QC
QC is a reference to the Scots Church (Presbyterian) in Pitt Street, Sydney
To confirm that this is the same person check the extract of the 1881 Census below...
The 1881 census was taken on the night of 3 April 1881 (http://www.findmypast.co.uk
COULTER, Robert John Head Married M 38 1843 Corporal R M L I Australia
COULTER, Annie Wife Married F 38 1843 Corporals Wife Stafford
Piece: 999 Folio: 73 Page: 10 Registration District: Eastry
Civil Parish: Walmer Address: Cavalry Barracks, Walmer Kent, Walmer
If you get his & Annie's marriage certificate it should confirm the fathers names...
FREE UK BMD (http://www.freebmd.org.uk/cgi/search.pl
Marriages Sep 1879
COULTER, Robert John & EMERY, Annie - RegD: Eastry, Vol: 2a, Page: 1300
Searchable Australian Convict Records are here http://www.convictrecords.com.au/
but David Coulter doesn't appear to be amongst them.
Don't underestimate the power of Google to help you out. Simply searching for the following string
"David Coulter" "Mary McNeice" provided links to information about Roberts parents...
(Apparently David Coulter was a grocer and died in 1876)
Robert's fathers death record tells us that Robert's gfather was a David Coulter too...
REG NO. LAST NAME GIVEN FATHER'S GIVEN MOTHER'S GIVEN DISTRICT
3366/1876 COULTER DAVID DAVID DIED PADDINGTON PADDINGTON
(unfortunately this record doesn't give us his mother's name)