Chromatin-based epigenetic memory relies on the accurate distribution of parental histone H3-H4 tetramers to newly replicated DNA strands. Mcm2, a subunit of the replicative helicase, and Dpb3/4, subunits of DNA polymerase ε, govern parental histone H3-H4 deposition to the lagging and leading strands, respectively. However, their contribution to epigenetic inheritance remains controversial. Here, using fission yeast heterochromatin inheritance systems that eliminate interference from initiation pathways, we show that a Mcm2 histone binding mutation severely disrupts heterochromatin inheritance, while mutations in Dpb3/4 cause only moderate defects. Surprisingly, simultaneous mutations of Mcm2 and Dpb3/4 stabilize heterochromatin inheritance. eSPAN (enrichment and sequencing of protein-associated nascent DNA) analyses confirmed the conservation of Mcm2 and Dpb3/4 functions in parental histone H3-H4 segregation, with their combined absence showing a more symmetric distribution of parental histone H3-H4 than either single mutation alone. Furthermore, the FACT histone chaperone regulates parental histone transfer to both strands and collaborates with Mcm2 and Dpb3/4 to maintain parental histone H3-H4 density and faithful heterochromatin inheritance. These results underscore the importance of both symmetric distribution of parental histones and their density at daughter strands for epigenetic inheritance and unveil distinctive properties of parental histone chaperones during DNA replication.