We needed a starting point to be able to help you and you've given it. So...
Son's Scottish birth certificate (c18886) will list date and place of parents' marriage. If the marriage took place in Ireland then you will get the residences of bride and groom at the time as well as their fathers' names & occupations which will probably allow you to get further back. Without more details like those you are unlikely to be able to search Irish records and be sure you've found the correct family.
Census records will also help. This looks like Joseph in Dalry, Ayrshire (I used to know it well) with wife and family. Since Janet was born in Scotland the marriage probably took place there.
1891- Bathbank, Dalry, Ayrshire, Scotland
Joseph Shillady, 34, Ireland, farm servant
wife Janet Shillady, 30, Mauchline Ayrshire, farm servant's wife
daughter Jane Shillady, 7, West Kilbride Ayrshire, scholar
son James Shillady,2, Dalry Ayrshire
Also 1 visitor (Elizabeth Denning, 13, West Kilbride, scholar) & 1 boarder
1901- East Barnaigh, Lochwinnoch, Renfrewshire
Joseph Shillady, 40, Ireland, farmer
wife Janet Shillady, 40, Mauchline Ayrshire
daughter Jane D. Shillady, 17, West Kilbride Ayrshire, farmer's daughter
son James Shillady, 12, Dalry Ayshire, scholar
daughter Janet Shillady, 6, Kilbirnie Ayrshire
Also 1 farm servant
Going back to 1881 census, this looks like Joseph:
1881- Meikle Busbie, Old Ardrossan, Ayrshire
servant Joseph Shelladay, 24, Ireland, farm servant
in household of Robert Brown