We consider the random series-parallel graph introduced by Hambly and Jordan (2004). The graph is built recursively: at each step, every edge in the graph is independently either replaced with probability $p$ by a series of two edges, or with probability $1-p$ by two parallel edges. At the $n$-th step of the recursive procedure, the distance between the extremal points on the graph is denoted by $D_n (p)$. It is known that $D_n(p)$ possesses a phase transition at $p=p_c :=\frac{1}{2}$ and we will study it's behavior in the slightly supercritical regime $p=p_c+\varepsilon$. Our main result says that as $\varepsilon\to 0^+$, the exponent $\alpha(p) := \lim_{n} \frac{\log E[D_n(p)]}{n} $ behaves like $\sqrt{\zeta(2) \, \varepsilon}$, where $\zeta(2) := \frac{\pi^2}{6}$.
