<p>Chiral materials exhibit a spin filtering effect, so-called chirality-induced spin selectivity (CISS). A recent observation of spin accumulation at the ends of a chiral-structured superconductor has opened up a new pathway for studying the CISS effect in superconductors. In chiral-structured superconductors, the admixture of the spin-singlet and spin-triplet order parameters significantly influences the properties of superconductivity. In this paper, we investigate the interplay between the superconducting order parameter and supercurrent-induced spin current, namely, the superconducting CISS effect. In weakly parity-mixed superconductors, the spin current, which is predominantly temperature-independent, is carried by spin-polarized Cooper pairs with finite center-of-mass momentum. In contrast, in strongly parity-mixed superconductors the temperature-dependent spin current is also carried by electrons with opposite momentum and antiparallel spins forming a Cooper pair. Our findings shed light on how parity mixing in the superconducting order parameter influences spin transport in crystals without inversion symmetry, notably those with chiral structures.</p>