The data for Catherine Fair-351 appears to just have the birth date off by 1 month.
Catherine Cockburn formerly Fair
Born 30 Nov 1813 in Ayton, Scotland
Daughter of Edward Fair and Elizabeth (Wood) Fair
Her date of birth should be 30 Dec 1813, but everything else is the same. The date of birth needs correcting, not the parents: What you are suggesting is essentially profile ID recycling. Regarding her birth date, it was probably a simple error of some kind or involving conflation... the two profiles could have been accidentally merged on FamilySearch (or another site) at one point, or it could be a typo. Hard to say, but your proposed remedy is wrong.
But the source data (which in this case says 30 Dec 1813) on FamilySearch is generally valid and trustworthy. It certainly is in this instance.
The same is true on ScotlandsPeople: Generally valid and trustworthy, although any index may contain or propagate errors. But that isn't the problem here, as you can see:
Again, there's an error in the date entered. Not an error in the parents (unless you have other, better evidence).
It appears that you're making some faulty assumptions. Don't be worried - everyone makes mistakes, including everyone here on WikiTree. The beauty of WikiTree is that through collaboration we have a feedback loop which can correct or undo those errors.