ABSTRACT We use particle-in-magnetohydrodynamics-cells to model particle acceleration and magnetic field amplification in a high-Mach, parallel shock in three dimensions and compare the result to 2D models. This allows us to determine whether 2D simulations can be relied upon to yield accurate results in terms of particle acceleration, magnetic field amplification, and the growth rate of instabilities. Our simulations show that the behaviour of the gas and the evolution of the instabilities are qualitatively similar for both the 2D and 3D models, with only minor quantitative differences that relate primarily to the growth speed of the instabilities. The main difference between 2D and 3D models can be found in the spectral energy distributions (SEDs) of the non-thermal particles. The 2D simulations prove to be more efficient, accelerating a larger fraction of the particles and achieving higher velocities. We conclude that, while 2D models are sufficient to investigate the instabilities in the gas, their results have to be treated with some caution when predicting the expected SED of a given shock.