The reflective property states that if a parabola can reflect light, then light that enters it travelling parallel to the axis of symmetry is reflected toward the focus.
(Taken from https://en.wikipedia.org/wiki/Parabola#/media/File:Parabel_2.svg)
Is parabola the unique curve with this property (i.e. "parallel to axis of symmetry" $\implies$ "reflected toward a single point") or are there other ones?
And is there a curve with a stronger property, namely any rays of light hitting the curve would be reflected toward a single point?
If you want a ray to be reflected at any point of the curve, then the curve must possess a tangent at every point, i.e. it must be represented by differentiable functions. (This immediately rules out the possibility that different rays incident at $P$ could be reflected at a same point).
Let's then set up a cartesian frame, having its origin at focus $F$ and $y$ axis parallel to the direction of the axis of symmetry of the curve. If $P=(x,y)$ is any point on the curve, then the reflective property implies that the tangent at $P$ is the bisector of angle $FPH$, where $PH$ is parallel to the axis. It follows that the tangent at $P$ meets the $y$ axis at a point $Q$ such that $FQ=FP=\sqrt{x^2+y^2}$. Hence we get for the derivative $y'(x)$, which is the slope of $PQ$, the following equation: $$ y'(x)={1\over x}\left(y+\sqrt{x^2+y^2}\right). $$ This can be solved to: $$ y={a}x^2-{1\over4a}, $$ where $a$ is an arbitrary integration constant. And this is indeed the equation of a parabola.