Standard time-stepping techniques for solving parabolic partial differential equations cannot be parallelized or vectorized efficiently unless the problem to be solved is very large and the number of processors is small. To overcome these limitations several researchers have presented methods that increase the parallel performance by calculating the solution on several time levels simultaneously. In earlier papers we have discussed such a method, based on a combination of the waveform relaxation method and multigrid. In this paper we extend our approach to the case of nonlinear parabolic partial differential equations. We illustrate the application of the new algorithm with numerical examples of initial-boundary-value and time-periodic type and we discuss its implementation on an Intel iPSC/2 hypercube. It is shown that also in the nonlinear case the multigrid waveform relaxation method outperforms the standard sequential solvers.