She committed the faux pas as the parliamentary education committee heard evidence from Audit Scotland and local authorities on the state of Scotland's schools.
Common flaws in new schools, auditors reported, related to light, ventilation and heating, prompting Ms Mulligan to announce proudly that the parliament building has no need for air conditioning.
Alistair Farquhar of Moray Council quickly replied that he was "determined not to comment on the suitability of the Scottish Parliament building".
Councils admitted, however, that, while their own efforts had perhaps come in on budget, schools' design had not always "set the world alight", to quote Lindsay Glasgow of Edinburgh City Council. But they were learning, it appeared. Public private partnership programme 2 had generally produced better schools than PPP1 and they expected improvement to continue, they said. In Moray, Mr Farquhar said, they were determined "not to make the same mistakes as, say, North Lanarkshire". They would make "entirely different ones".
When councils would next get the opportunity to allow their creative designer juices to flow was unclear. They needed Government support to build new schools, they said, but the SNP administration, having rejected PPPs and the Private Finance Initiative as "costly and flawed", had left "a vacuum".
Even a small authority such as Moray needed "lots of dosh", said Mr Farquhar. Its capital plan for education was pound;3 million a year, but an "all-singing, all-dancing" new secondary costs pound;30 million.
Edinburgh wants to build a new special school, three secondaries and a primary, costing pound;160 million. The council has pound;33 million.
Labour does not need to roll out its accounting guru, Arthur Midwinter, to show that the sums just don't add up.