Does the shortest distance between two curves exist along common normal?

Question

I read here that the shortest distance between two differentiable non-intersecting curves is along their common normal.

But if we consider $y^2=x+1$, $y^2 = x$, their common normal is actually the largest possible distance:

Maybe this is because the minimum distance between these two curves never exists in the first place... Is this a condition we should add in the statement?

Thanks