Host shifts, where a pathogen invades and establishes in a new host species, are a major source of emerging infectious diseases. They frequently occur between related host species and often rely on the pathogen evolving adaptations that increase their fitness in the novel host species. To investigate genetic changes in novel hosts, we experimentally evolved replicate lineages of an RNA virus (Drosophila C Virus) in 19 different species of Drosophilidae and deep sequenced the viral genomes. We found a strong pattern of parallel evolution, where viral lineages from the same host were genetically more similar to each other than to lineages from other host species. When we compared viruses that had evolved in different host species, we found that parallel genetic changes were more likely to occur if the two host species were closely related. This suggests that when a virus adapts to one host it might also become better adapted to closely related host species. This may explain in part why host shifts tend to occur between related species, and may mean that when a new pathogen appears in a given species, closely related species may become vulnerable to the new disease.