In this paper we study parabolically semihyperbolic generalized polynomial-like maps and give a finer fractal analysis of their Julia sets. We discuss various generalizations of the classical notion of topological pressure to situations in which the underlying potentials are not necessarily continuous or bounded. Subsequently, we investigate various types of conformal measures and invariant Gibbs states, which then enables us to deduce analytic properties for the generalized pressure functions. On the basis of these results, we finally derive our multifractal analysis, and then show that for the special case in which the Julia set does not contain critical points, this general multifractal analysis has a more transparent geometric interpretation in terms of the local scaling behaviour of the canonically associated equilibrium state.