Due to the rapid advancement of supercomputing resource, there is a growing interest in developing parallel algorithms for the large-scale reservoir simulation. In this paper, we present a parallel and fully implicit simulator for the black oil model based on the variational inequality (VI) framework, which can be used to enforce important mathematical and physical properties to obtain accurate constraint-preserving solutions. In other words, this framework ensures the predicted solution to stay within the physical range. In the proposed approach, the black oil model is reformulated as a variational inequality system that naturally satisfies the basic boundedness requirement of the solution, and then a fully implicit finite volume method is applied to discretize the model equations. In addition to that, a number of nonlinear and linear fast solver technologies, including a variant of inexact Newton methods and the domain decomposition based preconditioners, are employed to guarantee the robustness and parallel scalability of the simulator. A particular emphasis of the proposed framework is placed on the parallel and algorithmic performance of the variational inequality approach across large-scale and heterogeneous problems. Several numerical results pertaining to the problems in one, two and three dimensions are presented to illustrate the efficiency, robustness, and the overall performance of the fully implicit constraint-preserving simulator.