I suspect you've grasped at the right straw!
You know with certainty that your ancestor William Laurie died in 1870 and that his death certificate names his parents as Robert Laurie and Jean Scott. According to the 1851 census, William Laurie had just one son who was named Robert, indicating that William’s father was named Robert.
William’s age in the census indicates he was born about 1814. FamilySearch has a birth of William Lawrie at Carluke on 23 December 1814 with baptism 9 January 1814, parents Robert Lawrie and Jean Scott.
Births of children of Robert Lawrie and Jean Scott on FamilySearch are:
1794 Thomas Lawrie at Carluke
1796 Mary Lawrie at Carluke
1798 Martha Lawrie at Carluke
1800 James Brown Lawrie at Carluke
1803 Janet Lawrie at Carluke
1806 Jean Lawrie at Carluke
1808 Elizabeth Brown Lawrie at Carluke
1811 John Lawrie at Carluke
1813 William Lawrie at Carluke
The marriage of Robert Lawrie and Jean Scott therefore fits the one on 7 July 1792 at Carluke.
This also fits the 1841 census entry of:
Lauries Hall, Carluke, Lanarkshire
Robert Laurie labourer age 60
Jean Laurie age 65
William Laurie age 25
Elizabeth Brown age 25
(Elizabeth could have been their daughter Elizabeth Brown Lawrie born 1808)
So now you are looking for the birth and parents of Robert Laurie. From the 1841 census Robert would have been born between 1776 and 1781, and according to the burial record you found of Robert Lawrie at Carluke in 1843 his birth would have been in 1770 or 1771. But neither of those ages may have been accurate. His eldest son was named Thomas, which could indicate the likely first name of Robert’s father.
FamilySearch has Robert Lowrie baptised 1 December 1771 to Thomas Lowrie and Mary Forrest at Carstairs. Carstairs isn’t far from Carluke, and this is the only entry that looks feasible, so either it’s the right Robert or Robert’s baptism isn’t recorded.
Thomas Lawrie and Mary Forrest had the following children baptised:
1762 Robert Laurie at Lanark
1764 Janet Laurie at Lanark
1766 James Laurie at Lanark
1767 Anna Laurie at Lanark
1769 Margaret Laurie at Lanark
1771 Robert Laurie at Carstairs
It looks as if their first son Robert died, and between 1769 to 1771 they had moved to Carstairs, which is a neighbouring parish to Lanark.
Difficult to say with certainty going back that far, but I’d say there’s a good chance Robert Lawrie’s parents were Thomas Lawrie and Mary Forrest.