Automatic identification of jump Markov systems (JMS) is known to be an important but difficult problem. In this work, we propose a new algorithm for the unsupervised estimation of parameters in a class of linear JMS called " conditionally Gaussian pairwise Markov switching models " (CGPMSMs), which extends the family of classic " conditionally Gaussian linear state-space models " (CGLSSMs). The method makes use of the specific structure of its sub model called " conditionally Gaussian observed Markov switching model " (CGOMSM). The algorithm proposed consists in applying two EM algorithms sequentially: the first one is used to estimate the parameters and switches of the discrete pairwise Markov chain (PMC), which is a part of CGOMSM. Once estimated, it is used to sample switches and then the second one, called switching EM, is used to estimate the parameters of the distribution driving hidden states given the observations and the switches. The entire algorithm is evaluated with respect to data simulated according to CGPMSMs, and comparisons with several supervised methods attest its good efficiency.