We calculate the spreading exponents and some geometrical properties of avalanches in a novel avalanche model of solar flares, closely built on Parker's physical picture of coronal heating by nanoflares. The model is based on an idealized representation of a coronal loop as a bundle of magnetic flux strands wrapping around one another, numerically implemented as an anisotropic cellular automaton. We demonstrate that the growth of avalanches in this model exhibits power-laws correlations that are numerically consistent with the behavior of a general class of statistical physical systems in the vicinity of a stationary critical point. This demonstrates that the model indeed operates in a self-organized critical regime. Moreover, we find that the frequency distribution of avalanche peak areas A assumes a power-law form f(A) A ^−α_A with an index α _A 2.45, in excellent agreement with observationally-inferred values.