Suppose we have a parabola defined by $y = a x^2$, with $a \gt 0$. Now we want to know its length between $x_1$ and $x_2$.
$\begin{align} L &= \int dl \\ &= \int \sqrt{dx^2 + dy^2} \end{align}$
$y = a x^2$
$dy = 2 a x dx$
$\begin{align} L &= \int \sqrt{dx^2+ (2 a x dx)^2} \\ &= \int dx \sqrt{1+ (2 a x)^2} \end{align}$
Now we look in a table of integral rules:
$\int \sqrt{1 + (kx)^2} = \frac{1}{k} \sqrt{1 + k^2x^2} + \frac{1}{2} \ln\left(kx + \sqrt{1 + k^2x^2}\right) + C$
Now comes the hard part: given x1, x2 and L, solve for a. This looks like a trancendental equation and offhand I don't see any clever way to solve it analytically. So you'll have to solve it numerically. A simple loop of code should do it; L increases strictly with a.
Shifting to the desired x1, x2, y1 and y2 is trivial (but notice that the problem appears to be underconstrained).