This is a sequel of previous work by the authors [Ergodic theory Dyn. Syst. 20, No. 5, 1423-1447 (2000; Zbl 0982.37045)] that described and studied the invariant set \(J\) of a parabolic conformal iterated function system \(S\). A remarkable result in this paper gives full account of the behavior of the Hausdorff and packing measures of \(J\) in its Hausdorff dimension \(h\), for finite \(S\) in dimension \(d\geq 2\) satisfying the strong open set condition. It turns out that there is a trichotomy defined by the critical value \(h=1\). Also, it is proved that \(h\) coincides with the upper box-counting dimension of \(J\). Two independent sections are devoted to the study of dynamics of parabolic conformal diffeomorphisms in \({\mathbb R}^d\), \(d\geq 3\), and in dimension \(d=2\).