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We present a new optimization-based structure-preserving model order reduction (MOR) method for port-Hamiltonian descriptor systems (pH-DAEs) with differentiation index two. Our method is based on a novel parameterization that allows us to represent any linear time-invariant pH-DAE with a minimal number of parameters, which makes it well-suited to model reduction. We propose two algorithms which directly optimize the parameters of a reduced model to approximate a given large-scale model with respect to either the H-infinity or the H-2 norm. This approach has several benefits. Our parameterization ensures that the reduced model is again a pH-DAE system and enables a compact representation of the algebraic part of the large-scale model, which in projection-based methods often requires a more involved treatment. The direct optimization is entirely based on transfer function evaluations of the large-scale model and is therefore independent of the system matrices' structure. Numerical experiments are conducted to illustrate the high accuracy and small reduced model orders in comparison to other structure-preserving MOR methods.