Hi there,
1843 ish, NSW, Early Church Records, Lynne or Tynne or could it have been mis-transcribed as COYNE ? Remember, the index was not actually prepared until the 1930s, so there's lots of reasons for mistakes in the indexes (including of course, lack of records, lack of pages in a register, and not just poor handwriting in the registers, or errors in reading and transcribing those registers).
There’s an 1843 baptism indexed at NSW BDM for Ellen COYNE, Vol 121A, line 946, with parents as Edward and Mary.
Vol 121 is one of the Supplementary Registers
http://www.records.nsw.gov.au/state-archives/guides-and-finding-aids then
http://www.records.nsw.gov.au/state-archives/guides-and-finding-aids/short-guide-4/short-guide-4then
http://www.records.nsw.gov.au/state-archives/guides-and-finding-aids/short-guide-4/volumes-1-123-1/volumes-1-123 then Vol 121 (1-end (8789)) Supplementary Register Baptisms 1830-1901 is at Reel 5046
“transcriptions of baptismal entries copied by the Registrar General from church registers acquired or made available on loan from the various churches for check with official records”
Have you searched familysearch for a baptism in New South Wales 1843ish for Ellen or Eleanor or variations?
There may be elusive blanks in civil registration for NSW marriages 1856-1895. Some of those blanks include parents names

so of course, that info will be on the original Church registers.
Do we know the denomination for the family? (Which section of the cemetery ??)
Cheers JM