In this work, global sensitivity analysis (GSA) was used in conjunction with polynomial chaos expansion (PCE) to help assess unsaturated soil hydraulic parameters from experimental drainage data. The temporal sensitivity of different outputs (cumulative outflow, water content, and pressure head at different positions inside the column) to the Mualem–van Genuchten parameters was analyzed. In addition, the marginal effects of the parameters on the different outputs were analyzed across their entire uncertainty range. The results of the GSA revealed the necessary data set for a successful calibration of the hydraulic parameters. Parameter estimation was then performed in a Bayesian framework using a Markov chain Monte Carlo sampler with the PCEs instead of the discretized Richards’ equation model. The obtained parameters were validated by comparing the whole set of measurements to the results of the calibrated numerical model.