Wow was that poor Jane married to William Cox, widowed & left with 11 children by 1881? I see your dilemma married at 18 Nov. 28th 1860 Sydney Mines (St. Joseph's, North Sydney, Cape Breton )* no parents listed and so will not be with parents on the 1861 either!
In 1881 Jane b. 1842 W with children all irish R.C.: Laura Jane 19, William Robert 18 (Engineer ),
Richard Augustine 16 ( Engineer ), Alonza 14, ( birth says Alonzo Douglas) Rebecca Matilda 12, Mary Marget (* birth says Mary maria) 10,
John James 8, Charles 7, Frederick Robert (* birth says Robert Frederick) 5, Michael Montague 3,
Joseph FRANCIS 1 ...some different names here, might hold clues...
http://www.collectionscanada.gc.ca/databases/census-1881/001049-119.01-e.php?id=NS_534883The name Robert is mentioned twice in children's names...There is a Robert Hamilton b. 1809 with wife Alice in same district 1881. He is a Trader and born Ireland, Roman Catholic...might be worth looking into!
http://data2.collectionscanada.gc.ca/e/e325/e008115797.jpgalso an A.G. Hamilton in same district, a collector b. 1836 right next door
(baptism of Charles Francis Hamilton 1865 gives parents Alexander Hamilton & Hariet Mchardy
They are the only Irish Hamilton in Cape Breton, the rest are Scottish heritage
*http://search.labs.familysearch.org
http://search.labs.familysearch.org/recordsearch/start.html#p=collectionDetails&c=fs%3A1392507