Assuming the parabola is of the form $y = a(x-b)^2 + c$, the fact that $y(8) = y(16)$ tells you that the vertex of the parabola is halfway between $x=8$ and $x=16$; that is, at $x=12$. So $b=12$.
The slope of the parabola at the point $x=k$ is given by $2a(k-b)$ (this follows by taking the derivative: $y' = 2a(x-b)$, so evaluating at $x=k$ gives the quantity I just gave you). If you know $k$, then since we now know $b$ we can obtain the value of $a$. That will tells us the value of $a$ and $b$, and we only need to figure out the value of $c$. Since we know the value of $y$ at specific points, we can plug them into the formula and solve for $c$.
Once you know $a$, $b$, and $c$, you know the parabola.