These William's details?
William PEARSON b. 19 June 1851 and chr. 17 Aug. 1851 in Newburgh, Fife.
Parents: John PEARSON and Catharine REEDIE
From what you have, I don't think parents' married. Sometimes after an illegitimate birth (never mind being chastised by the Kirk Elders!) couples did go on to marry, but not sure that is what happened here.
If we have the right entry for Catherine, William's mother, I wonder if this is her in 1861 with a few more children:
Catherine Reddie 28, Linen Weaver b. Newburgh
Margaret Laing 2, daughter b. Brechin, Forfarshire
George Thomson 5, son b. Apper (?Upper) Clunie, Perthshire
Address: 7 Wallace Lane, Brechin Burgh, Angus
From this entry, if the right one, guessing Catherine is still unmarried and the children also illegitimate like William?
A few years later, there is this marriage showing:
CATTHREN REEDIE and JOHN LAING married on 31 Dec. 1866 in Brechin, Angus.
Given the date, this may show in 1867 on SP. Also watch spelling of Catherine. The above entry is from the IGI index and may well be how SP have indexed it.
Parents' details for this marriage will help you confirm whether this is the Catherine Reddie from Newburgh with the parents we likely already have.
Monica