The Voronoi diagram and its dual, the Delaunay triangulation, are fundamental structures in computational geometry, so it is natural to seek efficient parallel algorithms in special site configurations. This paper obtains a nearly work-optimal logarithmic-time algorithm for the Voronoi diagram of the vertices of a convex polygon. It works in O ( log n ) and uses processors in the concurrent-read concurrent write (CRCW) parallel random access machine (PRAM) model. The authors also obtain a nearly optimal logarithmic-time algorithm for the Voronoi diagram of the edges of a convex polygon operating within the same asymptotic resource bound. They believe that their methods can be extended to derive nearly optimal fast parallel algorithms for the more general site configurations for which linear-time and almost-linear-time sequential algorithms are known.