Data-driven discovery of partial differential equations (PDEs) has emerged as a promising approach for deriving underlying physics when domain knowledge about observed data is limited. Despite recent progress, the identification of governing equations and their parametric dependencies using conventional information criteria remains challenging in noisy situations, as the criteria tend to select overly complex PDEs. We introduce an extension of the uncertainty-penalized Bayesian information criterion (UBIC), which is adapted to solve parametric PDE discovery problems efficiently without computationally expensive PDE simulations. This extended UBIC uses quantified PDE uncertainty, accumulated across temporal or spatial points, to prevent overfitting in model selection. Numerical experiments on canonical PDEs show that our extended UBIC can identify the true number of terms and their varying coefficients accurately, even in the presence of noise.