The pursuit of sustainable management of Earth's resources through clean energy initiatives necessitates the precision offered by numerical simulations, which are often compromised by inherent uncertainties in model inputs and configurations. This study addresses the computational intensity of Bayesian inverse methods, particularly Markov Chain Monte Carlo (MCMC) techniques, by implementing Model Order Reduction (MOR). MOR significantly reduces the computational demand of forward model evaluations necessary for effective MCMC sampling, thus enhancing the feasibility of these methods in practical applications. However, the integration of MOR introduces approximation errors that could potentially skew the outcomes of the Bayesian inversion processes. Our research rigorously evaluates the impacts of these errors on uncertainty quantification within Bayesian frameworks, aiming to fortify the reliability of such methods in decision-making scenarios. The application of these methodologies to a coupled Thermo-Hydro (TH) problem characterized by complex thermal and hydrological interactions demonstrates the potential of MOR to streamline simulations and improve the accuracy and efficiency of uncertainty assessments in geoscientific models. This work not only advances the theoretical foundations of Bayesian inversions but also sets a precedent for their application in addressing critical challenges in the geosciences, particularly in energy and resource management.