Pure spin current, exhibiting no Joule heat and self-powered characteristics, has recently attracted intensive attention. Here, through first-principles calculations and symmetry analysis, we propose a new method to generate photoelectric pure spin current in carbon hexagonal connected three zigzag graphene nanoribbons (ZGNRs) via magnetic field modulation. Specifically, a device with centro-symmetry is designed, which consists of three ZGNRs using two carbon hexagons as connectors ('2-C6'). When the edge spin states of the three ZGNRs from left to right are modulated to AFM-AFM-AFM or FM-AFM-FM by magnetic fields, excellent pure spin currents are obtained which are independent of the photon energy and the angle of the linearly polarized light. However, when the edge spin states are FM-FM-FM orderly, the photocurrent is nearly zero and can be neglected. Analysis show that the first two spin magnetic structures own the spatial inversion antisymmetric spin density which is the origin of stable pure spin currents, while the FM-FM-FM structure owns Cs symmetric spin density, leading to the nearly zero photocurrent. Our findings provide a scheme for obtaining pure spin currents by changing the spin states of the graphene nanoribbons via magnetic field modulation, which is of great importance for the design of spintronic devices.