High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional gridbased methods tend to be frustrated by the curse of dimensionality. In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs the combination of reformulations in terms of backward stochastic differential equations and regression-type methods in the tensor format holds the promise of leveraging latent low-rank structures enabling both compression and efficient computation. Following this paradigm, we develop novel iterative schemes, involving either explicit and fast or implicit and accurate updates. We demonstrate in a number of examples that our methods achieve a favorable trade-off between accuracy and computational efficiency in comparison with state-of-the-art neural network based approaches.