arXiv:2610.11747v1 Announce Type: new Abstract: We propose and analyze two second-order basis-update \& Galerkin (BUG) time integration methods for dynamical low-rank approximation on Tucker tensors and tree tensor networks. Both are built from a sequential pre-computation sweep, after which the differential equations for all basis matrices and connecting tensors are evolved by a Galerkin method. Further, the proposed methods are rank-adaptive by construction. The first, the second-order parallel BUG integrator, solves all these differential equations fully in parallel, which is favorable on parallel architectures, followed by a sequential augmentation and truncation step. The second, the second-order augmented BUG integrator, gives up full parallelism but conserves norm and energy for Schr\"odinger equations and dissipates energy for gradient flows up to the truncation tolerance. For both integrators, we prove a second-order error bound that is robust with respect to small singular values of the matricization of connecting tensors. Numerical experiments for radiative transfer and quantum spin systems validate the theoretical findings.