We study a (1+ϵ)-approximate single-source shortest paths (henceforth, (1+ϵ)-SSSP) in n-vertex undirected, weighted graphs in the parallel (PRAM) model of computation. A randomized algorithm with polylogarithmic time and slightly super-linear work Õ(|E|•n^ρ), for an arbitrarily small ρ>0, was given by Cohen (10) more than 25 years ago. Exciting progress on this problem was achieved in recent years (4, 17, 19, 35), culminating in randomized polylogarithmic time and Õ(|E|) work. However, the question of whether there exists a deterministic counterpart of Cohen's algorithm remained wide open. In the current paper we devise the first deterministic polylogarithmic-time algorithm for this fundamental problem, with work Õ(|E|•n^ρ), for an arbitrarily small ρ>0. This result is based on the first efficient deterministic parallel algorithm for building hopsets, which we devise in this paper.