The points on the parabolas where the tangents have gradient 1 are on , and on . The distance between these points is .
In general, if you can't use that sort of symmetry to simplify the problem, you would have to find the equation of the normal at a general point on one curve. Then find the point(s), if any, where that line meets the other curve, determine the distance along the normal to the closest point on the other curve, and finally minimise that distance.
Edit. For the general case, Mr F's method (see next comment) is much better than my laborious procedure of using normals.