A large variety of nonlinear systems can be approximated by parallel Wiener-Hammerstein models. These models consist of a multiple input multiple output (MIMO) nonlinear static block sandwiched between two linear dynamic blocks. One method is available for the identification of a general parallel Wiener-Hammerstein model. It represents the nonlinear block as a multivariate polynomial, which typically contains cross-terms. These make it harder to interpret and to invert the model.We want to eliminate the cross-terms, and thus come to a decoupled polynomial representation. In this paper, the simultaneous decoupling of quadratic and cubic polynomials is formulated as a standard tensor decomposition. A simulation example shows that the simultaneous decoupling can result in a model with less parallel branches than a decoupling of all polynomials separately.