Notice that these are inverse functions of each other, you can swap x, y to get to the second parabola.
They are mirror images with respect to line x = y.
Required point should have this slope y′ = 1 for its tangent at point of tangency at ends of common normal.
Take the parabola with its symmetry axis coinciding with axis.
⇒ y2 = x − 1
Differentiating w.r.t x we get,
and the other point of tangency is again swapped to
Now use distance formula between them to get the minimum distance