  The Cantor locus is the unique hyperbolic component, in the moduli space of
quadratic rational maps ${\bf rat}_2$, consisting of maps with totally
disconnected Julia sets. Whereas the Cantor locus is well understood, its
boundary and dynamical relation to the boundary is not. In this paper we
explore the boundary of the Cantor locus near parabolic boundary points. We do
this by constructing a local parametrization, which fixes the dynamical
positions of the critical values, relative to the parabolic basin of the
quadratic polynomial $\mathrm{P}_{\omega_{p/q}}(z)=\omega_{p/q} z+z^2$,
$\omega_{p/q}=e^{2\pi ip/q}$. We characterize which pairs, of dynamical
positions and parabolic boundary points, give rise to bounded and unbounded
sequences in ${\bf rat}_2$ respectively. Moreover, we show that this approach
controls the dynamical postions of the critical values, for any limit point in
the bounded case, and for any limit point of a rescaled sequence of $q$th
iterates in the unbounded case.
