Model-based methods in autonomous driving and advanced driving assistance gain importance in research and development due to their potential to contribute to higher road safety. Parameters of vehicle models, however, are hard to identify precisely or they can change quickly depending on the driving conditions. In this paper, we address the problem of safe trajectory planning under parametric model uncertainties motivated by automotive applications. We use the generalised polynomial chaos expansions for efficient nonlinear uncertainty propagation and distributionally robust inequalities for chance constraints approximation. Inspired by the tube-based model predictive control, an ancillary feedback controller is used to control the deviations of stochastic modes from the nominal solution, and therefore, decrease the variance. Our approach allows reducing conservatism related to nonlinear uncertainty propagation while guaranteeing constraints satisfaction with a high probability. The performance is demonstrated on the example of a trajectory optimisation problem for a simplified vehicle model with uncertain parameters.