We develop a rigorous mathematical model of aqueous mineral carbonation kinetics for carbon capture and utilization (CCU) and estimate the parameter posterior distribution using Bayesian parameter estimation framework and lab-scale experiments. We conduct 16 experiments according to the orthogonal array design and an additional one experiment for the model test. The model considers the gas-liquid mass transfer, solid dissolution, ionic reactions, precipitations, and discrete events in the form of differential algebraic equations (DAEs). The Bayesian parameter estimation framework, which we distribute as a toolbox (https://github.com/jihyunbak/BayesChemEng), involves surrogate models, Markov chain Monte Carlo (MCMC) with tempering, global optimization, and various analysis tools. The obtained parameter distributions reflect the uncertain or multimodal natures of the parameters due to the incompleteness of the model and the experiments. They are used to earn stochastic model responses which show good fits with the experimental results. The fitting errors of all the 16 data sets and the unseen test set are measured to be comparable or lower than when deterministic optimization methods are used. The developed model is then applied to find out the operating conditions which increase the duration of high CO2 removal rate and the carbonate production rate. They have highly nonlinear relationships with design variables such as the amounts of CaCO3 and NaOH, flue gas flow rate, and CO2 inlet concentration.