A graph $\Gamma$ is edge-transitive ($s$-arc-transitive, respectively) if its full automorphism group $\rm Aut\,(\Gamma)$ acts transitively on the set of edges (the set of $s$-arcs in $\Gamma$ for an integer $s\geq 0$, respectively). A $1$-arc-transitive graph is called an arc-transitive graph or a symmetric graph. In this paper, we construct cubic symmetric bi-Cayley graphs over some groups of order $p^4$, where $p\geq 7$ is a prime. Using these constructions, we classify the connected cubic edge-transitive graphs of order $2p^4$ for each prime $p$ and we also show that all these graphs are symmetric.