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The paper "Enumeration and limit laws of series-parallel graphs" by Manuel Bodirsky, Omer Gimenez, Mihyun Kang, and Marc Noy, establishes that the number of labeled series-parallel graphs on $n$ vertices is asymptotically $$ g \cdot n^{-\frac{5}{2}} \gamma^n n! $$ where $g$ and $\gamma$ are constants. Perhaps you can convert their bound to one for unlabeled graphs in terms of edges.