A methodology for non-intrusive, projection-based nonlinear model reduction originally presented by Renganathan et. al. (2018) is further extended towards parametric systems with focus on application to aerospace design. Specifically, we extend the method for static systems with parametric geometry (that deforms the mesh), in addition to parametric boundary conditions. The main idea is to first perform a transformation on the governing equations such that it is lifted to a higher dimensional but linear underdetermined system. This enables one to extract the system matrices easily compared to that of the original nonlinear system. The underdetermined system is closed with a set of model-dependent nonlinear constraints upon which the model reduction is finally performed. The methodology is validated on the subsonic and transonic inviscid flow past the NACA0012 and the RAE2822 airfoils with parametrized shapes. We further demonstrate the utility of the approach by applying it to two common problems in aerospace design namely, derivative-free global optimization and parametric uncertainty quantification with Monte Carlo sampling. Overall, the methodology is shown to achieve accuracy upto 5% and computational speed-up of 2-3 orders of magnitude as that of the full-order model. Comparison against another non-intrusive model reduction method revealed that the proposed approach is more robust, accurate and retains the consistency between the state variables.