This article describes a numerical method based on the dual reciprocity boundary element method (DRBEM) for solving some well-known nonlinear parabolic partial differential equations (PDEs). The equations include the classic and generalized Fisher’s equations, Allen–Cahn equation, Newell–Whithead equation, FitzHugh–Nagumo equation, and generalized FitzHugh–Nagumo equation with time-dependent coefficients. The concept of the dual reciprocity is used to convert the domain integral to the boundary that leads to an integration-free method. We employ the time stepping scheme to approximate the time derivative, and the linear radial basis functions (RBFs) are used as approximate functions in the presented method. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in some numerical test examples. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.