Banning peanut products is an ineffective measure and probably violates the rights of other students and staff.
The ADA requires schools to make reasonable accomodations for students with disabilities. It does not require fellow students to make such accomodations. Banning peanut products from a school entirely places an undue burden on students who have limited financial means to purchase other food products, have food aversions or even just strong likes/dislikes of certain foods, have nutritional needs that require a high calorie, high protein diet, or who lack the time and ability to research their food choices and plan their lunch around the "needs" of someone not living in their own household.
It is also highly ineffective in preventing exposure, since as evidenced by this article, some students/families will fail to comply, either out of frustration, or by mistake, bringing an item that they don't realize contains peanuts.
Far better for the safety of the allergic student and the sanity of everyone else to provide a peanut-free area for the student and selected friends to eat their meal, while following accepted sanitation practices to minimize the likelihood of inadvertent exposure. Notice I didn't say, "prevent inadvertent exposure" because that is not entirely possible. When humans are involved, human error will occur.
The final step in this process is to develop a comprehensive health plan for this student, with input from the healthcare provider, parents, student, school nurse, education staff, lunch room workers, janitorial employees, etc., and educate everyone involved on how to prevent exposure, recognize exposure if it occurs, and respond swiftly to minimize effects.
I understand how frightening a severe allergy can be to parents, but they do themselves and their children NO favors by insisting on overly-intrusive, unnecessary and ineffective measures when there are better ways to address their children's needs.