Existing numerical schemes, maybe high-order accurate, are obtained on uniformly spaced meshes and challenges to achieve high accuracy in the presence of singular perturbation parameter, and nonlinearity remains left on nonuniformly spaced meshes. A new scheme is proposed for nonlinear 2D parabolic partial differential equations (PDEs) that attain fourth-order accuracy in $xy$-space and second-order exact in the temporal direction for uniform and nonuniform mesh step-size. The method proclaims a compact character using nine-point single-cell finite-difference discretization on a nonuniformly spaced spatial mesh point. A description of splitting compact operator form to the convection-dominated equation is obtained for implementing alternating direction implicit scheme. The procedure is examined for consistency and stability. The scheme is applied to linear and nonlinear 2D parabolic equations: convection-diffusion equations, Burger’s-Huxley, Burger’s-Fisher and coupled Burger’s equation. The technique yields the tridiagonal matrix and computed by the Thomas algorithm. Numerical simulations with linear and nonlinear problems corroborate the theoretical outcome.