Assessing seismic performance of existing structures after strong earthquake events is essential to identify potentially unsafe structures,and to schedule repairs or retrofitting. For this purpose, a well calibrated model of the structure of interest should be established based on a very limited number of seismic response data, by considering not only epistemic uncertainty due to lack of knowledge on actual mechanical properties of the structure but also aleatory uncertainty due to the manufacturing process and aging deterioration. In this work, a Bayesian model updating framework is proposed, where uncertainty characteristics of model predictions and observed data, both described in time domain, are quantified by the Bhattacharyya distance-based metric. This framework is employed for calibrating the probability distribution of stiffness parameters of a seismic-isolated bridge pier model by only five observed data reflecting aging condition of the isolators. The calibrated model is then employed to perform a seismic reliability analysis and is revealed that the failure probability of the target structure is increased due to aging of the isolators compared with the health condition, demonstrating that the proposed model updating procedure enables to quantitatively assess the seismic performance of the aging isolated bridges.