How to find the coefficient a of a y=ax^{2} parabola?

Step 1
As a first qualitative step we know that if the parabola opens upward
${\displaystyle a>0}$ otherwise if it opens downward ${\displaystyle a<0}$
For a quantitative evaluation we need at least to know the coordinates of a point ${\displaystyle (x_0,y_0)}$ on the parabola with ${\displaystyle x_0\ne 0}$ and then
${\displaystyle y_0=a{x}_0^2\Rightarrow a=\frac{y_0}{x_0^2}}$

Hold your parabolic mirror in a vertical position, with some distant object in front of it (the Sun, a tall tree with the sky as background, and so on) and a vertical white screen where to project the image.
Move your mirror until the image formed on the screen is clear and detailed. Measure then the distance between the image on the screen and the center of your mirror: that is the focal distance f of your screen. And
${\displaystyle a=\frac{1}{4f}}$