Is it permissible for a teacher to agree with a colleague to cover her classes for a financial compensation, with the knowledge of the school administration and teachers, but without the knowledge of the Ministry of Education, due to the teacher's need to care for her bedridden mother?

A teacher is a private employee. Therefore, it is not permissible for her to appoint someone else to teach in her place unless she has the permission of the employer. If the school administration is authorized to permit such a deputization, then there is no objection. Otherwise, the permission of the Ministry is necessary.