We consider reinforcement learning in parameterized Markov Decision Processes (MDPs), where the parameterization may induce correlation across transition probabilities or rewards. Consequently, observing a particular state transition might yield useful information about other, unobserved, parts of the MDP. We present a version of Thompson sampling for parameterized reinforcement learning problems, and derive a frequentist regret bound for priors over general parameter spaces. The result shows that the number of instants where suboptimal actions are chosen scales logarithmically with time, with high probability. It holds for priors without any additional, specific closed-form structure such as conjugate or product-form priors. Moreover, the constant factor in the logarithmic scaling encodes the information complexity of learning the MDP, in terms of the Kullback-Leibler geometry of the parameter space.