This paper presents an efficient method for extracting the second-order sensitivities from a system of implicit nonlinear equations. We design a custom automatic differentiation (AutoDiff) backend that targets highly parallel graphics processing units (GPUs) by extracting the second-order information in batch. When the nonlinear equations are associated to a reduced space optimization problem, we leverage the parallel reverse-mode accumulation in a batched adjoint-adjoint algorithm to compute efficiently the reduced Hessian of the problem. We apply the method to extract the reduced Hessian associated to the balance equations of a power network, and show that a parallel GPU implementation leads to a 30 times speed-up on the largest instances, comparing to our reference CPU implementation.