If she died in Scotland after 1855, her death certificate should tell you the names of her parents.
I see that there were at least five children
John, born 27 March 1856, Abbey, died 1856, Abbey
John, born 6 February 1858, Abbey
Margaret, born 24 April 1862, Dalry, died 1869, Dalry
Thomas, born 19 April 1863, Dalry
William, born 3 May 1866, Dalry
(The Dalry in Ayrshire)
In 1871 the family are at 1871 North Street, Dalry: John, 43; Margaret, 36; Thomas, 8; and William, 5. Both John and Margaret born in Johnstone, Renfrewshire
In 1861, there's a household in Smith Street, Dalry: John, 32; Margaret, 33; Samuel, 13; Mary, 8; James, 6; Jane, 3; Thomas Gibson, 27; his wife Elizabeth, 29; sons Samuel, 6 and John, 4. All born in Johnston, Renfrewshire
In 1881 John, 55; Margaret, 56; and William, 14 are at New Street, Dalry.
Margaret Gibson or McAulay, aged 78, died in Ardrossan in 1900.
Go to
www.scotlandspeople.gov.uk, invest in a few credits, and use 6 of them to view and download her death certificate.