  We say a measure is $C^{1,\alpha}$ $d$-rectifiable if there is a countable
union of $C^{1,\alpha}$ $d$-surfaces whose complement has measure zero. We
provide sufficient conditions for a Radon measure in $\mathbb{R}^n$ to be
$C^{1,\alpha}$ $d$-rectifiable, with $\alpha \in (0,1]$. The conditions involve
a Bishop-Jones type square function and all statements are quantitative in that
the $C^{1,\alpha}$ constants depend on such a function. Along the way we also
give sufficient conditions for $C^{1,\alpha}$ parametrizations for Reifenberg
flat sets in terms of the same square function. Key tools for the proof come
from David and Toro's Reifenberg parametrizations of sets with holes in the
H\"{o}lder and Lipschitz categories.
