  We study the dynamics of kinks in the $\phi^4$ model subjected to a
parametric ac force, both with and without damping, as a paradigm of solitary
waves with internal modes. By using a collective coordinate approach, we find
that the parametric force has a non-parametric effect on the kink motion.
Specifically, we find that the internal mode leads to a resonance for
frequencies of the parametric driving close to its own frequency, in which case
the energy of the system grows as well as the width of the kink. These
predictions of the collective coordinate theory are verified by numerical
simulations of the full partial differential equation. We finally compare this
kind of resonance with that obtained for non-parametric ac forces and conclude
that the effect of ac drivings on solitary waves with internal modes is exactly
the opposite of their character in the partial differential equation.
