Gaussian mixture models are universal approximators in the sense that any smooth density can be approximated arbitrarily well with a Gaussian mixture model with enough components. Due to their broad expressive power, Gaussian mixture models appear in many applications. As a result, algebraic parameter recovery for Gaussian mixture models from data is a valuable contribution to multiple fields. Our work documents performance of the method of moments for high dimensional Gaussian mixtures. We outline the method of moments, and selections of moments and their corresponding polynomials that work well for parameter recovery in practice. Our main contribution puts these ideas into practice with an implementation as a julia package, GMMParameterEstimation, as well as computational benchmarks.