This paper examines a model involving two dynamic Erdős-Rényi random graphs that evolve in parallel, with edges in each graph alternating between being present and absent according to specified on- and off-time distributions. A key feature of our setup is regime switching: the graph that is observed at any given moment depends on the state of an underlying background process, which is modeled as an alternating renewal process. This modeling framework captures a common situation in various real-world applications, where the observed network is influenced by a (typically unobservable) background process. Such scenarios arise, for example, in economics, communication networks, and biological systems. In our setup we only have access to aggregate quantities such as the number of active edges or the counts of specific subgraphs (such as stars or complete graphs) in the observed graph; importantly, we do not observe the mode. The objective is to estimate the on- and off-time distributions of the edges in each of the two dynamic Erdős-Rényi random graphs, as well as the distribution of time spent in each of the two modes. By employing parametric models for the on- and off-times and the background process, we develop a method of moments approach to estimate the relevant parameters. Experimental evaluations are conducted to demonstrate the effectiveness of the proposed method in recovering these parameters.