I would tell you to stand your ground...but I know this type of mom. Your best bet is to get with the VP and tell them it's bottom band or no band, and if they are willing, let them be the one to drop the hammer on it.
Also, don't e-mail this parent. All communications need to be over the phone or in person with a witness. E-mails will be saved and picked over for nuggets they can cling to.
Make sure your admin has your back. If they don't , this parent can make your life miserable.