In 2.98 and later you can add a Mirror modifier to a Bezier curve and use a Hook modifier and an empty if you want the mirror point to be one of the end points of the Curve.
- Create your Bezier curve.
- Move the 3D cursor to the end of the curve you want to be the mirror point.
- Add an empty. It will appear at the mirror point.
- Edit the Bezier curve and select the end point.
- Add a hook modifier, set it to the empty, and assign it.
- Add a Mirror modifier, assign the Mirror Object field to the empty, and adjust axis as you'd like.
Here's a simple Bezier curve, mirrored on its left end:
Here's the modifier stack
You have to remember that when you want to move that endpoint you do it by moving the empty. Otherwise you can treat the original exactly as you would otherwise.