  We show that the infinite-dimensional versions of Gowers' $\mathrm{FIN}_k$
and $\mathrm{FIN}_{\pm k}$ theorems can be parametrized by an infinite sequence
of perfect subsets of $2^\omega$. To do so, we use ultra-Ramsey theory to
obtain exact and approximate versions of a result which combines elements from
both Gowers' theorems and the Hales-Jewett theorem. As a consequence, we obtain
a parametrized version of Gowers' $c_0$ theorem.
