The incident happened at Brechin High School on Monday.
A slide show containing personal details of around 50 pupils was screened in a pre-prepared presentation.
Conditions included ADHD and autism.
Angus Council has launched an investigation.
The presentation detailed how exam arrangements for children with additional support needs would be different, and listed individual pupils.
The child's mother told the BBC: "You're just shocked, because these are teachers, they're educators.
"They do data protection training courses on a regular basis.
"In this age of confidentiality and data protection how would you think it was reasonable to do that?"
"I won't be leaving this here and I don't think it should just be swept under the carpet with an internal investigation.
"My concern now is how these vulnerable young people are going to be supported."
Angus Council said the school's head teacher was contacting the parents of the pupils whose details were shared.
A spokesman from Angus Council said: "We apologise for the obvious upset and concern this has caused, particularly to those young people whose details were shown.
"Inquiries are under way to establish the full circumstances of this isolated incident and whether any individual learning requires to be provided."