Realizability and parametricity are two well-known approaches to the semantics of System F, the architectural language for polymorphism. Many well-known realizability semantics can be recast in a simple topological form as induced by closure operators over sets of lambda-terms. This allows to generalize some completeness results known in the literature to a wide class of semantics (including Krivine's saturated sets and several variants of Girard's reducibility candidates), and to relate realizability with parametricity and dinaturality, an approach to parametricity arising from the functorial semantics of polymorphism. Our main result is that for a general class of realizability semantics (those which satisfy a particular topological property) one can prove a parametricity theorem'' stating that closed realizers are parametric and adinaturality theorem'' stating that closed realizers of positive types are dinatural. We compare our results with Wadler's approach which sees realizability and parametricity as some sort of adjoint functors. Finally, we briefly discuss the case of Girard's original definition of reducibility candidates, whose ``not trivial and somehow mysterious'' [Riba 2007] structure does not fit yet within our approach.