The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. We propose the first optimal deterministic BSP algorithm for computing the convex hull of a set of points in three-dimensional Euclidean space. Our algorithm is basedon known fundamental results from combinatorial geometry, concerning small-sized, efficiently constructible ϵ-nets and ϵ-approximations of a given point set. The algorithm generalises the technique of regular sampling, usedpreviously for sorting andt wodimensional convex hull computation. The cost of the simple algorithm is optimal only for extremely large inputs; we show how to reduce the requiredinput size by applying regular sampling in a multi-level fashion.