By extending and nonlinearly coupling two one-dimensional parabolic discrete maps, a new two-dimensional parabolic discrete map is achieved. By using stability analysis of fixed points and bifurcation analysis of map, the complex dynamical behavior and attractor evolution of the proposed two-dimensional discrete map are investigated, and its peculiar dynamical characteristics, such as the coexisting bifurcation modes and fast-slow periodic oscillation effects, etc., are illustrated. The research results indicate that two-dimensional parabola discrete map has two control parameters with different functions of adjustable dynamical behaviors and adjustable dynamic amplitudes, and there emerge nonlinear physical phenomena of Hopf bifurcation, bifurcation mode coexisting, locked-frequency and periodic oscillation fast-slow effect. Furthermore, the corresponding theoretical analysis and numerical simulation results are verified based on a digital circuit realized by microcontroller.