When used in engineering applications, most existing chaotic systems may have many disadvantages, including discontinuous chaotic parameter ranges, lack of robust chaos, and easy occurrence of chaos degradation. In this article, we propose a two-dimensional (2-D) parametric polynomial chaotic system (2D-PPCS) as a general system that can yield many 2-D chaotic maps with different exponent coefficient settings. The 2D-PPCS initializes two parametric polynomials and then applies modular chaotification to the polynomials. Setting different control parameters allows the 2D-PPCS to customize its Lyapunov exponents in order to obtain robust chaos and behaviors with desired complexity. Our theoretical analysis demonstrates the robust chaotic behavior of the 2D-PPCS. Two illustrative examples are provided and tested based on numeral experiments to verify the effectiveness of the 2D-PPCS. A chaos-based pseudorandom number generator is also developed to illustrate the applications of the 2D-PPCS. The experimental results demonstrate that these examples of the 2D-PPCS can achieve robust and desired chaos, have better performance, and generate higher randomness pseudorandom numbers than some representative 2-D chaotic maps.