N.H. Rev. Stat. Ann. § 194:28

If any town in which a high school or school of corresponding grade is not maintained neglects or refuses to pay for tuition as provided in RSA 194:27, such town shall be liable therefor to the parent or guardian of the child furnished with such tuition, if the parent or guardian has paid the same; otherwise to the town or city furnishing the same in an action of assumpsit.