The New York City Department of Education has only received about 72,000 -- or 20% -- of the consent forms that were sent out for children returning to the classroom at the beginning of the school year, ABC New York station WABC reported.
However, some students have been tested anyway, according to the station.
Several parents informed assembly member Mike Reilly, who represents South Shore of Staten Island, of the incidents, Reilly wrote on Facebook last week, adding that the Special Commissioner of Investigation "is looking into the matter."
One mother, Tina Angilletta, told WABC she felt her parental rights were violated.
"I didn't agree to anything. I didn't consent to anything" Angellitta said. "So in that sense, I feel like my parental rights were violated."
The DOE is taking "additional steps" to ensure that students who have not been given permission to be tested do not receive one, such as making sure staff is using the most up-to-date consent lists, according to a statement.
"This should have never happened and we are adjusting our protocols to ensure it does not happen again," DOE Deputy Press Secretary Nathaniel Styer said.
Experts are still unsure as to whether schools could serve as super-spreader sites for the virus.
The testing "is a critical part of keeping in-person learning safe and healthy for students and staff," Styer said.
Currently, there are no "red zone" areas in Staten Island, an interactive map by the city’s Department of Information Technology and Telecommunications shows.
Neighborhoods in Queens and Brooklyn contain micro-clusters of the virus, as well as Orange and Rockland Counties outside the city.
Despite the hot spots, New York has one of the lowest infection rates in the nation at a 1.1% average as of Saturday, Cuomo said.
The city's DOE did not provide additional comment to ABC News.