Peanut/tree nut is one of the allergy categories covered on the restaurants that offer allergy menus on mobile order, so assuming the place you want to eat does have an allergy menu, then your dad will be covered. If not, you can order at the register.
It's easy to see in the My Disney Experience app which places have allergy menus - just start the ordering process (you won't have to finish and pay). If there's an allergy category, it will be at the bottom of the menu.
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