We provide geometric sufficient conditions for Reifenberg flat sets of any integer dimension in Euclidean space to be parametrized by a Lipschitz map with Hölder derivatives. The conditions use a Jones type square function and all statements are quantitative in that the Hölder and Lipschitz constants of the parametrizations depend on such a function. We use these results to prove sufficient conditions for higher order rectifiability of sets and measures. Key tools for the proof come from Guy David and Tatiana Toro’s parametrization of Reifenberg flat sets in the Hölder and Lipschitz categories. If time allows, we will discuss some related work in progress and an example that shows that the conditions are not necessary.