The development of methods for generating pure spin current at the molecular level is vital. In this work, we investigated how the spin-related photocurrent is produced in a cobalt phthalocyanine chain by the photogalvanic effect (PGE). Depending on how the magnetic moments of the left and right halves of the cobalt phthalocyanine chain are arranged, spin current can be generated. Both charge current and spin current are absent when the magnetic moments are arranged in parallel. Pure spin currents are generated when the magnetic moments are arranged in an antiparallel manner. Importantly, the pure spin current is robust to the polarization type and polarization angle. This characteristic results from the structure's charge density having spatial inversion symmetry but lacking that of the spin density.