Fusiliers Dougald McCaughey, 23, 17-year-old John McCaig and his brother Joseph McCaig, 18, were lured from a Belfast bar by the IRA in 1971 on the pretext of meeting girls. They were then taken to north Belfast and shot.
It was the first time the IRA targeted off-duty soldiers, causing tens of thousands to take the streets in protest.
Nobody has ever stood trial for the murders.
Mr Mason made headlines when he was asked via Twitter to support the campaign for a private prosecution against the killers.
He responded that he was “not taking sides between Irish and British” and said that some people refer to the IRA as “Irish freedom fighters”.
Mr Mason has remained defiant over the past week and refused to retract his comments.
However after persistent requests by the Belfast News Letter newspaper to SNP leader Nicola Sturgeon and former leader Alex Salmond for an apology and other media coverage of the matter, the SNP issued an apology on Tuesday afternoon on behalf of Mr Mason, to the News Letter.
He said: “I deeply regret the offence and upset that has been caused to the relatives who lost loved ones and am extremely sorry that this has happened.
“I condemn and deplore all acts of terrorism. I do not intend to make any further comment.”
David McCaughey, a cousin of two of the soldiers, said: “He would have been better not to have got involved in the first place. Why did he just not say ‘no comment’.
“This apology now is a bit like closing the door after the horse has bolted.
“No doubt Nicola Sturgeon and Alex Salmond told him to do it because they just don’t want the grief this is causing the party.”
Campaign director of The Three Scottish Soldiers, Kris McGurk, said: “I welcome the apology from John Mason and the Scottish National Party on behalf of the Families and Campaign.
“I will be emailing John tonight to let him know the campaign team and families accept his apology. We all feel that if the apology is heartfelt and sincere, which we hope it to be, then this is the correct thing for us to do.
“I would now hope that as an MSP who has members of the McCaughey family living within his constituency, John Mason will offer us his full support.
“Myself, [our lawyers] McCue and Partners and both families have a long and busy road ahead of us, I am glad this issue is now resolved so we can concentrate on the real important issue and goal here which is justice.”
• A version of this article newsletter.co.uk