A rapidly forced parametric perturbation of sine-Gordon equation is considered. Using the normal form reduction it is shown that the system is well approximated by the double sine-Gordon equation (that is, with $\sin 2\phi$ instead of $\sin\phi$). The reduced equation possesses $\pi$-kinks, which are expected to be present in the original system. Some physical consequences of this result for an idealized ferromagnetic model are described.