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Technically it isn't. The ADA requires that an employer grant reasonable accommodations in the workplace for your disabilities, not those of your family members.
What it does prohibit though is discrimination against you based on your relationship with someone that has a disability. So, the only way the ADA would help here is if the employer allows scheduling flexibility with other employees based on their needs, but refuses to grant the same flexibility to you simply because your request is based on your child having a disability. That really narrows the possible infractions here though. If the employer is generally inflexible with the schedule, then you can't really make the argument that the employer is just targeting you simply because your child is disabled.
On the facts that you've outlined here, the employer is correct that there is no obligation to switch your shift.