This paper presents an optimal reduced-dimension Kalman filter for a family of triplet Markov models (TMM). The problem is to estimate the state vector in the case when the auxiliary process in the TMM can be eliminated. Sufficient conditions for this elimination to be feasible are established and we give a selection of illustrative real-life TMM examples, where these conditions are satisfied. We subsequently show that the original TMM boils down to a pairwise Markov model (PMM) of second order. Then, we derive a new optimal Kalman filter applicable to any linear PMM of second order. Our numerical results confirm that the proposed estimator can provide substantial complexity reduction with either no or minor accuracy loss, depending on the use of model approximation.