All-optical phase regeneration aims at restoring the phase information of coherently encoded data signals directly in the optical domain so as to compensate for phase distortions caused by transceiver imperfections and nonlinear impairments along the transmission link. Although it was proposed two decades ago, all-optical phase regeneration has not been seen in realistic networks to date, mainly because this technique entails complex bulk modules and relies on high-precision phase sensitive nonlinear dynamics, both of which are adverse to field deployment. Here, we demonstrate a new, to the best of our knowledge, architecture to implement all-optical phase regeneration using integrated photonic devices. In particular, we realize quadrature phase quantization by exploring the phase-sensitive parametric wave mixing within on-chip silicon waveguides, while multiple coherent pump laser tones are provided by a chip-scale micro-cavity Kerr frequency comb. Multi-channel all-optical phase regeneration is experimentally demonstrated for 40 Gbps QPSK data, achieving the best SNR improvement of more than 6 dB. Our study showcases a promising avenue to enable the practical implementation of all-optical phase regeneration in realistic long-distance fiber transmission networks.