Nuclear spin diffusion in double quantum wells (QWs) is examined by using dynamic nuclear polarization (DNP) at a Landau level filling factor v = 2/3 spin phase transition (SPT). The longitudinal resistance increases during the DNP of one of the two QW (the "polarization QW") by means of a large applied current and starts to decrease just after the termination of the DNP. On the other hand, the longitudinal resistance of the other QW (the "detection QW") continuously increases for approximately 2 h after the termination of the DNP of the polarization QW. It is therefore concluded that the nuclear spins diffuse from the polarization QW to the detection QW. The time evolution of the longitudinal resistance of the polarization QW is explained mainly by the nuclear spin diffusion in the in-plane direction. In contrast, that of the detection QW manifests much slower nuclear diffusion in the perpendicular direction through the AlGaAs barrier.