Flow-accelerated corrosion (FAC) is a life-limiting factor for the piping network of the primary heat transport system (PHTS) in CANDU® reactors. The pipe wall thinning caused by FAC is monitored by carrying out periodic in-service inspections (ISI) to ensure the fitness-for-service of the piping system. Accurate prediction of the lifetime of various components in the PHTS piping network requires estimation of FAC thinning rate. The traditional Bayesian inference techniques commonly employed for parameter estimation are computationally costly. This paper presents an inexpensive and intuitive simulation-based Bayesian approach to FAC rate estimation, called approximate Bayesian computation using Markov chain Monte Carlo (ABC-MCMC). ABC-MCMC is a likelihood-free Bayesian computation scheme that generates samples directly from an approximate posterior distribution by simulating data sets from a forward model. The efficiency of ABC-MCMC is demonstrated by presenting a comparison with a likelihood-based Bayesian computation scheme, Metropolis-Hastings (MH) algorithm, using a practical data-based example. Furthermore, an innovative step has been proposed for reducing the Markov chain burn-in time in the proposed scheme. To indicate the need of a Bayesian approach in quantifying the uncertainties related to the FAC model parameters, results from the linear regression method, a common industrial approach, are also presented in this study. The numerical results show a notable reduction in computational time, suggesting that ABC-MCMC is an efficient alternative to the traditional Bayesian inference methods, specifically for handling noisy degradation data.