In this paper we study the $\sigma$-Jarnik limit sets ${\cal J}_{\sigma}$ of geometrically finite Kleinian groups with parabolic elements. By generalizing the methode of Jarnik in the theory of Diophantine approximations, we estimate the dimension of ${\cal J}_{\sigma}$ with respect to the Patterson measure. In the case `$\delta \leq k_{max}$, which includes all finitely generated Fuchsian groups, it is shown that this leads to a complete description of ${\cal J}_{\sigma}(G)$ in terms of Hausdorff dimension. For the remaining case we derive some estimates for the Hausdorff dimension as well as the packing dimension of ${\cal J}_{\sigma}(G)$.