We study parallel sampling from high-dimensional strongly log-concave distributions. Langevin-based samplers converge rapidly in continuous time, but their discretizations are typically sequential and often require polynomially many steps in the dimension $d$, the target accuracy $\varepsilon^{-1}$, or both. Picard-based parallel sampling methods reduce this sequential depth to polylogarithmic scale by solving for many time-discretization points in parallel; however, existing guarantees often require a polynomial number of processors, leading to substantial memory and gradient-evaluation costs in high dimensions. We show that higher-order Langevin structure can reduce this parallel resource burden while preserving polylogarithmic sequential depth. Our method combines arbitrary-order Langevin dynamics with blockwise Lagrange polynomial interpolation. This sharper discretization reduces the number of parallel points required to achieve a target accuracy. Our results cover both higher-order smooth potentials and ridge-separable potentials, including models such as Bayesian logistic regression and two-layer neural networks, and improve upon the space complexity of the current literature on parallel log-concave sampling.