Alice Parker appears in the 1860 US Census as born about 1841 in Canada.
There is no Alice Parker listed in the 1851 Canada Census for the family. There is a woman who appears to be named Elishun, but who is born about 1834 in Ireland.
RootsIreland.ie has a baptismal record for an Ellen Parker for 27 Dec 1832.
My current hypothesis is these records all reference the same person.