In this paper we consider rational functions $f:\oc \to \oc$ with parabolic and critical points contained in their Julia sets $J(f)$ such that $\sum_{n=1}^\infty|(f^n)'(f(c))|^{-1}<\infty$ for each critical point $c \in J(f)$. We calculate the Hausdorff dimensions of subsets of $J(f)$ consisting of elements $z$ for which $\inf\{\dist(f^n(z),\Crit(f)):n\ge 0\}>0$ and which are well-approximable by backward iterates of the parabolic periodic points of $f$.