How parental histone (H3-H4)2 tetramers, the primary carriers of epigenetic modifications, are transferred onto leading and lagging strands of DNA replication forks for epigenetic inheritance remains elusive. Here we show that parental (H3-H4)2 tetramers are assembled into nucleosomes onto both leading and lagging strands, with a slight preference for lagging strands. The lagging-strand preference increases markedly in budding yeast cells lacking Dpb3 and Dpb4, two subunits of the leading strand DNA polymerase, Pol ε, owing to the impairment of parental (H3-H4)2 transfer to leading strands. Dpb3-Dpb4 binds H3-H4 in vitro and participates in the inheritance of heterochromatin. These results indicate that different proteins facilitate the transfer of parental (H3-H4)2 onto leading versus lagging strands and that Dbp3-Dpb4 plays an important role in this poorly understood process.