If you were to redraw half the shape, tracing the original, you can then use the Reflect Tool and click the Copy Button. Then merely join the two halves.
If it's already a vector shape, merely cut it in half, reflect, and rejoin to make the sides symmetrical.
To the best of my knowledge, there's no mechanism to symmetrically edit more than one existing anchor point, or existing Bezier handle. There are perhaps some third-party plug-ins which used to do that (i.e. Better Handles), but I don't know if they still exist for CC versions.