In this report, we define a sound and complete categorical semantics for the parallel lambda-calculus, based on a notion of aggregation monad which is modular w.r.t. associativity, commutativity and idempotence. To prove completeness, we introduce a category of partial equivalence relations adapted to parallelism, in which any extension of the basic equational theory of the calculus is induced by some model. We also present abstract methods to construct models of the parallel lambda-calculus in categories where particular equations have solutions, such as the category of Scott domains and its variants, and check that G. Boudol's original semantics is a particular case of ours.