Hornby High principal Richard Edmundson confirmed that is the case.
He refused to comment further.
"We want to give the boy privacy. It's a matter between the student, his family, and the school," Mr Edmundson said.
Katrina Casey, the ministry's head of sector enablement and support, said the ministry has been in touch with the school.
"They've told us they are working with the family to work though the issue," she said.
"The mother has been in touch with us and we have provided her with some advice."
The ministry said it was satisfied the school is making necessary arrangements to encourage the student back to school.
The school had also made arrangements for the student to sit an assessment that was due.
"We would encourage the mother to continue talking to the school and work through this issue so her son can return to school," Ms Casey said.
"There are also some other schools that could be suitable for her son if she wants him to attend another school, that do not have enrolment zones."
A spokesman for Ms Parata confirmed that she had received a letter from Ms Peebles.
"However, this is not a matter [Ms Parata] will be commenting on," he said.
"School rules and the enforcement of those rules are matters for school boards and school management."
Boards of trustees are responsible for setting school rules on a wide range of matters, including those on uniform and appearance.
The ministry said facial hair was not grounds to stand down or suspend a student.
However, if a student repeatedly refuses to comply with school rules, that could amount to "continual disobedience".
"For this to result in a stand-down or suspension, the continual disobedience must be a harmful or dangerous example to other students at the school," the ministry says.
"In making any rules, such as those setting out a prohibition on facial hair, the board would be required to consider other laws such as those relating to human rights.
"For example there may be cases where exceptions are made in relation to facial hair to take into account a student's genuinely held religious beliefs."
Ms Peebles could not be reached for comment.