Parameter estimation for complex models using Bayesian inference is usually a very costly process as it requires a large number of solves of the forward problem. We propose here an approach to reduce the computational cost by constructing surrogate models that provide approximations of the true solutions of the forward problem. The surrogate models are built in an adaptive manner using a posteriori error estimates for quantities of interest in order to control their accuracy. Effectiveness of the proposed methodology is demonstrated on a numerical example dealing with the parameter calibration of the Spalart-Allmaras model for the simulation of turbulent channel flows.