The paper extends Moschovakis higher-order type theory of acyclic recursion by adding type polymorphism. We extend the type system of the theory to model parametric information that pertains to underspecified types. Different kinds of type polymorphism are presented via type variables and recursion constructs for alternative, disjunctive type assignments. Based on the new type system, we extend the reduction calculus of the theory of acyclic recursion. We motivate the type polymorphism with examples from English language.