An Independent Parallelism Theorem is proven in the theory of adhesive HLR categories. It shows the bijective correspondence between sequential independent and parallel independent direct derivations in the Weak Double-Pushout framework, see [2]. The parallel derivations are expressed by means of Parallel Coherent Transformations (PCTs), hence without assuming the existence of coproducts compatible with M as in the standard Parallelism Theorem. It is aslo shown that a derived rule can be extracted from any PCT, in the sense that to any direct derivation of this rule corresponds a valid PCT.