The algebraic methods for serial and parallel decompositions of Mueller matrices are combined in order to obtain a general framework for a suitable analysis of polarimetric measurements based on equivalent systems constituted by simple components. A general procedure for the parallel decomposition of a Mueller matrix into a convex sum of pure elements is presented and applied to the two canonical forms of depolarizing Mueller matrices [Ossikovski, J. Opt. Soc. Am. A 27, 123 (2010).], leading to the serial-parallel decomposition of any Mueller matrix. The resultant model is consistent with the mathematical structure and the reciprocity properties of Mueller matrices.