An important open problem in the synthesis of passive controllers is to obtain a passive network that realizes an arbitrary given impedance function and contains the least possible number of elements. This problem has its origins in electric circuit theory, and is directly applicable to the cost-effective design of mechanical systems containing the inerter. Despite a rich history, the problem can only be considered solved for networks that contain at most two energy storage elements, and in a small number of other special cases. In this article, we solve the minimal network realization problem for the class of impedances realized by series-parallel networks containing at most three energy storage elements. To accomplish this, we develop a novel continuity-based approach to eliminate redundant elements from a network.