The uncertainty in the probabilistic assignment of the system parameters can strongly affect the reliability assessment of engineering systems. As such, it should be properly accounted for at the design stage. Imprecise probability descriptions have been investigated for establishing bounds on the failure probability. However, the computational burden associated with the propagation of the imprecise probability description is often too high, making this type of robust reliability analyses impractical. A machine learning optimization strategy is presented to reduce the computational cost of the propagation of imprecise probability descriptions obtained when considering probability and cumulative density functions with non-probabilistic parameters (i.e. interval and convex). This type of description is employed, for example, in Parameterized P-Boxes and in the Generalized Maximum Entropy distribution under uncertain statistical information. The proposed approach evaluates the Lower Bound (LB) and Upper Bound (UB) on the failure probability by adaptively selecting the values of the non-probabilistic parameters to yield two separate Gaussian Process Regression models: one for the UB and one for the LB. The proposed uncertainty propagation approach is applied to the evaluation of the failure probability bounds on the random vibration response of a structural panel with uncertainty in the probabilistic assignment of the system parameters, showing a drastic reduction in computational cost.