Is following true?
If a parabola and a branch of hyperbola are tangent at 2 points and on the same side of the common tangents, then the hyperbola is "outside" the parabola.
Example:
To generalize, two conics with $e\ge1$ are tangent at 2 points and on the same side of the common tangents, then the one with greater eccentricity lies on the "outside". (here is my calculation for a right hyperbola and a hyperbola)