  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 2-D models. This allows us to determine
whether 2-D 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 2-D and
3-D models, with only minor quantitative differences that relate primarily to
the growth speed of the instabilities. The main difference between 2-D and 3-D
models can be found in the spectral energy distributions (SEDs) of the
non-thermal particles. The 2-D simulations prove to be more efficient,
accelerating a larger fraction of the particles and achieving higher
velocities. We conclude that, while 2-D 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.
