<p>This paper develops a numerical method based on a combination of the Crank–Nicolson finite difference technique and two modified cubic B-spline differential quadrature methods for solving one- and two-dimensional parabolic Volterra partial integro-differential equations. This type of equation frequently arise in various scientific and engineering applications, making their accurate and efficient solution highly significant. Initially, the Crank–Nicolson finite difference method is utilized to discretize the time variable, while the trapezoidal integration rule is employed for the nonlinear Volterra integral component. Subsequently, the differential quadrature method, which uses two modified cubic B-spline basis functions, is applied to derive a fully discrete iterative scheme. The stability and convergence of the time-discretized scheme are rigorously analyzed using the energy method. Furthermore, the Richardson extrapolation technique is employed to enhance the order of convergence in the time direction. To demonstrate the accuracy and efficiency of the proposed method, several one- and two-dimensional examples are presented at the end of the paper.</p>