Altermagnetism has advanced the study of compensated magnets by revealing non-relativistic spin splitting, traditionally classified into strictly even- or odd-parity spin textures. Here, we unveil a fundamentally different regime: component-resolved mixed-parity spin splitting in a fully three-dimensional compensated magnet. Using first-principles calculations, tight-binding and k · p models, along with spin-group symmetry analysis, we demonstrate that the non-coplanar ground state of TbB₄ enforces a unique momentum-space spin texture. The in-plane spin components exhibit odd-parity p- and f-wave-like textures, whereas the out-of-plane component retains an even-parity d-wave altermagnetic character. Crucially, the coexistence of the in-plane odd-parity textures is driven not by relativistic spin-orbit coupling, but by a staggered Berry phase arising from the inherent scalar spin chirality. This mixed-parity structure dictates distinct transport fingerprints, including bulk non-relativistic Edelstein and spin Hall responses, as well as a symmetry-allowed Berry curvature dipole. These results establish the rare-earth tetraborides as a robust platform for engineering complex spin-charge conversion phenomena.