We study the h-conformal measure for parabolic rational maps, where h denotes
the Hausdorff dimension of the associated Julia sets. We derive a formula which
describes in a uniform way the scaling of this measure at arbitrary elements of the
Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational
maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian
groups) and show that this law provides some efficient control for the fluctuation of
the h-conformal measure. We then show that these results lead to some refinements
of the description of this measure in terms of Hausdorff and packing measures with
respect to some gauge functions. Also, we derive a simple proof of the fact that the
Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the
conformal measure is a regular doubling measure, we show that its Renyi dimension
and its information dimension are equal to h and we compute its logarithmic index.