In this research paper, a numerical method, named the three-step Ultraspherical wavelet collocation method, is presented for solving some nonlinear multi-dimensional parabolic partial differential equations. The method is third-order accurate in time. In this method, the three-step Taylor method is used to get the time derivative, while the Ultraspherical wavelet collocation method is used to get the space derivatives. Ultraspherical wavelets have good properties which make useful to carry out this aim. The presented method is developed for Burgers' equation, Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation, and quasilinear parabolic equation. Three illustrative numerical problems are solved to demonstrate the efficiency, simplicity, and reliability of the presented method.