A tree on Ancestry shows Elizabeth's parents as Charles & Ann nee Brough. It also shows Ann Robertson as a half sister to Elizabeth
Makes no sense as the Robertson/Brough marriage was 1833??
Well, it is Ancestry - I wouldn't expect it to make sense. But it's worth looking at.
According to the OPRs at SP, Charles R and Ann Brough had two sons, Charles in 1834 and Andrew in 1836.
The family are in Perth in 1841: Charles, 35; Ann, 30; Charles, 6; Andrew, 4; Mary, 3; James, 1; plus John Brown, 6 months and Esther Paul, 11.
In 1851 (surname spelled Roberson) there are Charles, 49; Ann, 36; Charles, 17; Andrew, 15; Mary, 13; Ann, 10; Cathrine, 7; James, 11; Jean, 4; and Margret, 1.
They are also in Perth in 1861: Ann Brouch, married, 46; Charles, 27; Margaret, 10; Elisbath, 8; and grandson Andrew Glasgow, 3. Charles senior wasn't with his wife and the youngest members of his family.
So there is an Ann, but if she was really 10 in 1851 she was born in 1840/1841, not 1843; and a
full sister Elizabeth, born 1852/1853. It's quite possible that this is your Ann Robertson.
As suggested, you really need to get Elizabeth's marriage certificate to find out her father's name and occupation.
Also take a look at that marriage in Edinburgh in 1863. In fact, do that first, because if it's her it will name both her parents, unlike Elizabeth's English certificate which will cost you more and tell you less.