This paper presents a novel Bayesian framework for time-domain operational modal analysis (BTOMA) inspired by machine learning time series prediction. The framework incorporates Bayesian regularization and multi-setup information fusion under unknown excitation, offering inherent parallelism for efficient uncertainty quantification of modal analysis. Key features of the proposed BTOMA include: (1) A measurement setup parallelism (SP) strategy that leverages modern parallel and distributed computing capabilities, significantly enhancing uncertainty quantification efficiency for ambient excitation analysis of large-scale civil engineering structures. (2) A likelihood function based on free vibration response prediction, coupled with parameterized likelihood and prior distributions for modal parameters. (3) An efficient joint inference strategy for modal parameters and regularized hyperparameters, utilizing a Bayesian learning framework based on nonlinear least-squares (NLSQ) and Gauss-Newton approximation for high-dimensional Hessian matrices. (4) A multi-setup information fusion approach that dramatically improves modal analysis efficiency for large-scale structures. The framework's effectiveness is validated through numerical studies on a multi-story shear frame model and a long-span suspension bridge model, as well as the Z24 bridge benchmark with multi-setup measurement data. Results demonstrate the BTOMA's potential for enhancing operational modal analysis in complex structural systems.