Let $A=\mathbf{Z}/4\mathbf{Z}$. Then $A$ is flat over itself, but the ideal $I:=2\mathbf{Z}/4\mathbf{Z}$ is not flat over $A$. This is because when we tensor the injection $I\hookrightarrow A$ with $I$, we get the map $I\otimes_AI\rightarrow I$ which sends $r\otimes s$ to $rs$, is visibly the zero map. The group $I\otimes_AI$ is not zero because it is isomorphic to $(\mathbf{Z}/2\mathbf{Z})\otimes_{\mathbf{Z}/2\mathbf{Z}}(\mathbf{Z}/2\mathbf{Z})\cong \mathbf{Z}/2\mathbf{Z}$. So $I\otimes_AI\rightarrow I$ is not injective.
However, if $A$ is a principal ideal domain, or more generally a Dedekind domain, then submodules of flat $A$-modules are flat, because for such a ring $A$, flat$=$torsion-free, and it is clear that submodules of torsion-free modules are torsion-free (over any domain).