It is acknowledged by mechanical and aeronautical engineers that many important industrial fluid mechanics simulation-based decisions are made by analysing Reynolds-Averaged Navier-Stokes (RANS) simulations. However, despite that the use of RANS simulations has become a necessity, still unrealistic conditions such as not considering experimental variability or blindly trust a turbulence model happens in practice. This is an important concern in simulation of high-speed jet flows, as the reliability and the accuracy of RANS is still an issue. In this work non-intrusive Uncertainty Quantification (UQ) has been applied to the linear stability analysis of 3D RANS simulations of an under-expanded supersonic jet under uncertainty. The Parabolized Stability Equations (PSE) are utilised for the first time in the literature in a probabilistic approach for the propagation of uncertainty upon CFD simulations. As PSE does not include turbulence nor stagnation pressure as parameters in their mathematical formulation (named non-intrinsic parameters in this work), this framework enables a recommended practice to quantify the impact of uncertainty from earlier stages on the latter jet instability analysis. For this objective, an artificial stochastic base-flow is generated from UQ on RANS to develop the final quantification of uncertainties on jet stability characteristics with PSE. Spatial uncertainty quantification results show that the considered stagnation pressure uncertainty (equivalent to mass-flow rate uncertainty) affects mainly to the shock-cells area, which is in agreement with the jet flow mechanics. The tested turbulent viscosity ratio uncertainty from the Spalart-Allmaras turbulence model has been observed more influential on the shear layer region in general, and with a special role on the growth rate of the perturbations. Frequency (in terms of Strouhal number), has been found influential on the spatial localisation of the turbulent wave packets. This procedure is proposed as an standard approach in the field. The framework may help to understand how instability features are being affected by expected or unexpected uncertainty sources and can guide to future robust developments in the industry, for instance in jet noise reduction.