Due to storage requirements and computational complexity, the approximate solution of PDEs computed by standard numerical methods is usually limited to problems with up to three or four dimensions. In our contribution, we use recently proposed wavelets based on quadratic splines to solve the Poisson equation with Dirichlet boundary conditions in higher dimensions. We describe here our algorithm of matrix-vector multiplication which takes account of the special structure of wavelet stiffness matrices and enables scalable parallelization. At the end, we show its performance for problems up to six dimensions.