  There are two ways to compute Poincar\'e-Dulac normal forms of systems of
ODEs. Under the original approach used by Poincar\'e the normalizing
transformation is explicitly computed. On each step, the normalizing procedure
requires the substitution of a polynomial to a series. Under the other
approach, a normal form is computed using Lie transformations. In this case,
the changes of coordinates are performed as actions of certain infinitesimal
generators. In both cases, on each step the homological equation is solved in
the vector space of polynomial vector fields $V^n_j$ where each component of
the vector field is a homogeneous polynomial of degree $j$. We present the
third way of computing normal forms of polynomial systems of ODEs where the
coefficients of all terms are parameters. Although we use Lie transforms the
homological equation is solved not in $V^n_j$ but in the vector space of
polynomial vector fields where each component is a homogeneous polynomial in
the parameters of the system. It is shown that the space of the parameters is a
kind of dual space and, the computation of normal forms can be performed in the
space of parameters treated as the space of generalized vector fields. The
approach provides a simple way to parallelize the normal form computations
opening the way to compute normal forms up to higher order than under
previously known two approaches.
