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The ADA generally applies to the public areas of schools. There are exceptions, including schools owned by religious entities that do not receive federal funding.
Public areas of schools generally include administrative offices, the gymnasium during a public sporting event, the auditorium during a public exhibition, or the cafeteria during a public fund raising pancake supper. The class rooms themselves are generally not considered public areas of the school and therefore would not be covered under Title III of the ADA.
If a parent wishes his or her child to attend public school with a service animal, the first step would be to add the service animal to the child's IEP (Individual Education Plan), under IDEA (Individuals with Disabilities Education Act). If the school is not willing to permit the service animal, the parent(s) can appeal to the Department of Education.
Colleges and universities (and other postsecondary education institutions such as vocational-technical schools) operate with considerable autonomy and with some state supervision. Therefore, you should start by contacting your state department of higher education to find out how they can help you resolve your problem.