The periodic motion of the interacting vortex domain walls in a pair of nanostripes has been theoretically investigated. As a model, two parallel nanostripes with magnetization inhomogeneities in the form of magnetic vortices have been examined. The magnetic subsystems of the stripes are magnetostatically coupled. The quasi-elastic coupling between vortices ensures the existence of normal modes of the periodic magnetization motion. The frequencies of these modes have been calculated. It is shown that not any combination of the vortex topological charge leads to the resonant behavior of magnetization in ac fields. The effect of the static component of a magnetic field on the frequency of the periodic motion of vortex domain walls is discussed.