Mr Peacock said he would raise these issues with council education conveners at a meeting planned for next month. But he cautioned: "We are not masters of everything in this respect. Local authorities are autonomous organisations.
"It is also important to remember that, under freedom of information legislation, people can get access to information and, if they choose to make results into league tables, there is little I can do about it."
Contacted later by The TES Scotland, Jim Fletcher, education convener in high-performing East Renfrewshire, confirmed that the authority would be publishing school-level exam tables this week. It is also considering how to extend awareness of the information to parents.
"In our regular meetings with school board chairs, there has never been any suggestion that parents do not want this information," Mr Fletcher said.
"But if we are told not to publish, we won't publish."
The Executive has decided to limit published exam results to individual local authority and national statistics. Parents will be able to access results for their child's school or for schools to which they want to send their children via a new website to be unveiled next month.
The site will include other performance data so schools can be judged "in context".