Momentum-dependent spin splitting and its relation to inversion (P) and time-reversal (T) symmetries are central to nonrelativistic spintronics. Representative examples include collinear altermagnets with (P,T)=(+,-) and non-collinear odd-parity magnets with (P,T)=(-,+). In this work, we develop a theoretical framework to induce odd-parity spin splittings in the more abundant collinear altermagnets through two mechanisms: driving by a two-color linearly polarized light field or coupling to a P-odd loop-current order. Properly phase-locked two-color driving induces a static (P,T)=(-,-) order, symmetry-equivalent to a translationally invariant P-odd loop-current order. Coupling this order to an altermagnet produces a controllable mixed-parity spin texture, opening new avenues for the electrical and optical manipulation of spin-polarized currents in spintronics applications. The same mechanism applied to a collinear PT-symmetric magnet induces a distinct (P,T)=(+,+) state with a nonrelativistic dissipationless anomalous spin Hall conductivity. We present group-theory and microscopic Floquet theory to highlight the emergent responses.