Can schools change classes of a child with an IEP without parental permission? What are a parent's legal rights in this?

If you are not satisfied with the way the school is handling your child’s needs, you have a right to contest it. An IEP is a legal document and you are a member of the IEP team. The following articles from LD OnLine have information about parental rights that might be useful to you:

If you need additional support, consult with a special education attorney or the school system’s legal counsel for answers to your specific question. It may be time to rewrite the IEP to better reflect your child’s current level of functioning.