Mimicking the parametrized Ellentuck theorem in the Ellentuck space and the parametrized Milliken theorem in the Milliken space, we present a 'moderately abstract' parametrized theorem for 'moderate' topological Ramsey spaces. It is a parametrization of the abstract Ellentuck theorem with infinitely many perfect sets of real numbers, implying that essentially all infinitely-dimensional Ramsey properties proven using topological Ramsey space theory can be parametrized by products of infinitely many perfect sets.