A bivariate Markov chain comprises a pair of random processes which are jointly, but not necessarily individually, Markov. We are interested in continuous-time finite-alphabet bivariate Markov chains. Only one of the two processes of the bivariate Markov chain is observable. The observable process, and the other underlying process, may jump simultaneously. Examples of bivariate Markov chains include the Markov modulated Poisson process, and the batch Markovian arrival process, when suitable modulo counts are applied. An Expectation-Maximization (EM) algorithm for maximum likelihood estimation of the parameter of a bivariate Markov chain was recently developed. Here we prove strong consistency of maximum likelihood parameter estimation for the bivariate Markov chain. We extend the proofs developed by Leroux for hidden Markov processes and by Rydén for Markov modulated Poisson processes, to bivariate Markov chains. Such chains do not generally have a hidden Markov process representation.