The class of parallel rewriting systems is considered in this work, and the interaction between two complexity measures, that in the literature have been called synchronous parallelism and independent parallelism, is investigated. It is shown that, when the degree of synchronous parallelism is bounded by some constant greater than one, the degree of independent parallelism induces an infinite non-collapsing hierarchy within the family of generated languages. The result is obtained using an original characterization of parallel rewriting systems. Our result combines with other well known properties of synchronous parallelism to reveal the existence of a two-dimensional hierarchy for the family of languages generated by so called finite copying parallel rewriting systems. This gives a new picture of many formalisms in this class. Other language-theoretic properties of parallel rewriting systems are proved in this work, that together with our main result provide an answer to some questions that were left open in the literature.