The study of tumor growth biology with computer-based models is currently an area of active research. Different simulation techniques can be used to describe the complexity of any real tumor behavior, among these, "cellular automata"-based simulations provide an accurate tumor growth graphical representation while, at the same time, keep simpler the implementation of the automata as computer programs. Several authors have recently published relevant proposals, based on the latter approach, to solve tumor growth representation problem through the development of some strategies for accelerating the simulation model. These strategies achieve computational performance of cellular-models representation by the appropriate selection of data types, and the clever use of supporting data structures. However, as of today, multithreaded processing techniques and multicore processors have not been used to program cellular growth models with generality. This paper presents a new model that incorporates parallel programming for multi and manycore processors, and implements any synchronization requirement necessary to implement the solution. The proposed parallel model has been proved using Java and C++ program implementations on two different platforms: chipset Intel i5-4440 and one node of 16-processors cluster of our university. The improvement resulting from the introduction of parallelism into the model is analyzed in this paper, comparing it with the standard sequential simulation model currently used by researchers in mathematical oncology.