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In the literature, there are quite a few sequential and parallel algorithms to solve problems on distance-hereditary graphs. Two well-known classes of graphs, which contain trees and cographs, belong to distance-hereditary graphs. We consider the vertex-coloring problem on distance-hereditary graphs. Let Td(|V|, |E|) and Pdd(|V|, |E|) denote the time and processor complexities, respectively, required to construct a decomposition tree representation of a distance-hereditary graph G=(V,E) on a PRAM model Md. Our algorithm runs in O(Td(|V|, |E|)+log|V|) time using O(Pd(|V|, |E|)+|V|/log|V|) processors on Md. The best known result for constructing a decomposition tree needs O(log2 |V|) time using O(|V|+|E|) processors on a CREW PRAM. If a decomposition tree is provided as input, we solve the problem in O(log |V|) time using O(|V|/log |V|) processors on an EREW PRAM. To the best of our knowledge, there is no parallel algorithm for this problem on distance-hereditary graphs.