We introduce a new construct that can be used to parametrize some topological Ramsey spaces by the collection of infinite subsets of the natural numbers. We show that these parametrized spaces are also topological Ramsey spaces. Then we use these spaces to give new proofs of some known parametrized perfect set theorems. We conclude with a discussion of how to extend the results to the abstract setting and open questions related to applying the results to the Tukey theory of ultrafilters.