We develop a novel parallel decomposition strategy for unweighted, undirected graphs, based on growing disjoint connected clusters from batches of centers progressively selected from yet uncovered nodes. With respect to similar previous decompositions, our strategy exercises a tighter control on both the number of clusters and their maximum radius. We present two important applications of our parallel graph decomposition: (1) $k$-center clustering approximation; and (2) diameter approximation. In both cases, we obtain algorithms which feature a polylogarithmic approximation factor and are amenable to a distributed implementation that is geared for massive (long-diameter) graphs. The total space needed for the computation is linear in the problem size, and the parallel depth is substantially sublinear in the diameter for graphs with low doubling dimension. To the best of our knowledge, ours are the first parallel approximations for these problems which achieve sub-diameter parallel time, for a relevant class of graphs, using only linear space. Besides the theoretical guarantees, our algorithms allow for a very simple implementation on clustered architectures: we report on extensive experiments which demonstrate their effectiveness and efficiency on large graphs as compared to alternative known approaches.