Estimating the sky background is critical for ground-based astronomical research. In the optical, scattered moonlight dominates the sky background, when the moon is above the horizon. The most uncertain component of a scattered moonlight model is the aerosol scattering. The current, official sky background model for Cerro Paranal uses an extrapolated aerosol extinction curve. With a set of X-shooter sky observations, we have tested the current sky background model as well as determined the aerosol extinction from the ultra-violet (UV) to near-infrared (NW). To our knowledge, this is the first time that a scattered moonlight model has been used for this purpose. These observations were taken of blank sky, during three different lunar phases, and at six different angular distances from the moon for each lunar phase. Overall, the current model does reproduce the observations for average conditions quite well. Using a set of sky background models with varying aerosol distributions to compare with the observations, we found the most likely aerosol extinction curves, phase functions, and volume densities for the three nights of observations and compare them with the current model. While there are some degeneracies in the aerosol scattering properties, the extinction curves tend to flatten towards redder wavelengths and are overall less steep compared to the extrapolated curve used in the current model. Also, the current model had significantly less coarse particles compared to the favored volume densities from the X-shooter data. Having more coarse particles affects the phase function by being more peaked at small angular distances. For the three nights of sky observations, the aerosol size distributions differed, most likely reflecting the changes in atmospheric conditions and aerosol content, which is expected. In short, the current sky background model is in fair agreement with the observations, and we have determined better aerosol extinction curves and phase functions for Cerro Paranal. Using nighttime sky observations of scattered moonlight and a set of sky background models is a new method to probe the aerosol content of the atmosphere.