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We describe a framework for the simultaneous estimation of model parameters in a partial differential equation using sparse observations. Markov Chain Monte Carlo sampling is used in a Bayesian framework to estimate posterior probability distributions for each parameter. We describe the necessary components of this approach and its broad potential for application in models of unsteady processes. The framework is applied to three case studies, of increasing complexity, from the field of cohesive sediment transport. We demonstrate that the framework can be used to recover posterior distributions for all parameters of interest and the results agree well with independent estimates (where available). We also demonstrate how the framework can be used to compare different model parameterizations and provide information on the covariance between model parameters. Plain Language Summary: We describe a framework for the simultaneous estimation of multiple unobserved parameters by combining observations of a tracer with a numerical model. This framework uses Bayesian inference techniques established in statistical literature to estimate the unobserved parameters of interest used in the model with uncertainty quantification. We explain the key components of this framework in simple terms to encourage its use for analyzing other unsteady processes and performing quantitative inference on parameters that are difficult or impossible to measure directly. We then demonstrate the framework's efficacy by applying it to three case studies from the field of cohesive sediment transport that all use the transport equation (advection‐diffusion). Inferred parameter values show good agreement with independent estimates, where available. Key Points: Probabilistic framework to estimate unobserved erosion model parameters using sparse measurements collected above the seabedGeneral approach can be updated with any model parameterization and quantitatively comparedThe framework is applicable to many similar data sets with both unsteady or quasi‐steady forcing and response [ABSTRACT FROM AUTHOR]