$p$-wave magnets exhibit odd-parity spin polarization in momentum space, with spin splitting that reverses under $\vec{k}\to -\vec{k}$, while preserving zero net magnetization. Here we show that, in odd-parity magnets with spin-orbit coupling, elliptically polarized light generates in-plane momentum-independent spin-dependent terms that dynamically switch a zero-net-magnetization $p$-wave state into a finite spin-polarized state. The Floquet-engineered bands can also acquire a nonzero Chern number, whose sign is controlled by the light polarization. For $f$-wave magnets, circularly polarized light eliminates the uniform in-plane component and leaves net out-of-plane magnetization originating from the spin-orbit coupling with polarized light. Our results establish light as an efficient means of controlling magnetic states, with potential applications in spintronics and quantum information.