The time to compute normal forms numerically grows very fast with respect to
the degree of the normal form and the dimension of the system. We design and
implement a parallel algorithm based on the parallel computation of Poisson
brackets. The implementation is based on previous work by A.~Jorba, and uses
the C programming language and PVM. We present a theoretical and empirical
efficiency analysis of the algorithm. We also suggest some possible
improvements of the algorithm.