We consider parameterization in o-minimal structures and look at how the o-minimal Reparameterization Theorem of Pila and Wilkie might be enhanced. In particular, we are interested in whether or not we can have any greater control over the bounds on the derivatives of the parameterizing functions. Work by Pila shows that a certain choice of bounds, namely `mild' bounds, could improve the original corollary to the Reparameterization Theorem (a result about the bound on the number of rational points of bounded height lying on definable sets), at least in the particular case of Pfaff curves. We consider in which o-minimal structures mild parameterization might be found and, using work of Le Gal, show that the analogous reparameterization theorem does not hold for o-minimal structures in general.