Nonlinear dynamical systems are very often subject to parametric uncertainties. Two challenges arise when studying the resulting nonlinear dynamics. First, the cost of running simulations of high-dimensional models can be prohibitive. Second, parametric uncertainties render deterministic simulations both inefficient and inconclusive. To tackle these challenges, we present an approach for parametric model reduction. We develop polynomial chaos expansions of low-dimensional invariant manifolds, known to exist by the theory of spectral submanifolds , in high-dimensional systems of ordinary differential equations. This enables us to construct efficient parametric reduced models. In addition, we derive closed-form expressions for response characteristics such as nonlinear backbone curves and damping ratios of nonlinear mechanical systems.