Pupils at the Vale First and Middle School, in Vale Avenue, Findon Valley, were performing a rendition of Bugsy when both Nol Atkins and his wife Helena dozed off during the first half.
Nol and his wife had been invited to attend the show as guests of honour and were sitting in the front row.
One parent, who did not want to be named, said: "I think it is appalling. The couple were the guests of honour but within 20 minutes they had both fallen asleep.
"It's just disappointing for the children. Some of them were very upset and started crying during the interval, and asked what they were doing wrong.
"They thought the mayor was bored because their performance wasn't good enough."
Another parent, the father of a year seven pupil, said: "It was just an example of poor manners. I wouldn't go to a West End show and start nodding off."
Last Wednesday, the day after the performance, Mr Atkins issued an apology.
He said: "First of all, I would like to apologise if I have upset any of the children at the Vale First and Middle School by nodding off during their performance.
"Unfortunately, my wife, Helena, who is recovering from a stroke, is on medication and sometimes this makes her drowsy. For myself, a heavy cold and a packed day did not help, not that this is an excuse.
"A reviving cup of coffee did work wonders, and we were pleased to meet and chat with the children during the interval and enjoy their performance of Bugsy."
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