Unfortunately the marriage certificate will not include details of the parents, and neither will Mary-Ann's death certificate. William's death certificate should name his place of birth and his parents, so long as the informant knew the details.
I had a quick look on Papers Past, and there are a few short articles about William's death in a workplace accident (e.g. https://paperspast.natlib.govt.nz/newspapers/GLOBE18820184.108.40.206). There seems a discrepancy between the age given there and in the cemetery records, which suggests he was born about 1829, and the birth date you have for him.
Likewise, Mary-Ann was born about 1841 according to her death notice in the newspaper, not 1836. Both dates make it unlikely she was born in Dunedin, which wasn't established until 1848. Unfortunately the early settlers database for Otago isn't online to check whether there is a Richards family settled in the area before 1848 (see http://www.toituosm.com/learn/research).
Wow, looks like William might have had an interesting background - I found the Hocken Library's index of early residents of Otago (https://www.otago.ac.nz/library/hocken/oni/basic.php) and looked up William Curtis. There's lots of entries from electoral rolls etc but one is more interesting: from the Police Gazette and reads 'Information required of a woman named Stewart; it is stated she went to New Zealand some time ago with a Cornish man named William Curtis; Victoria Police Gazette 3 Jan 1877 page 4'.
This suggests that William might be from Cornwall, England rather than Cornwall, Ontario. it's much more likely that he was born in England than in Canada.
Separating out people of the same name is tough at that period. Have you had a look at the passenger lists online, to see if you can find a William Curtis born 1829/30 and Mary-Ann Richards? https://www.familysearch.org/search/collection/1609792 and https://freepages.rootsweb.com/~nzbound/genealogy/lists.htm are the biggest collection but there are others at FamilySearch too.