If two parabolas don't intersect, is there a line that doesn't intersect either of them? Two parabolas in a plane are given, such that they don't intersect. Is it true that there is a line in plane such that doesn't intersect any of them?
Parabolas are convex, so they'll always be on the same side of their tangent lines. Find both tangent lines at the points of closest approach. These lines are parallel. Intuitively, if they were not then taking a 'small step' along the parabola in the right direction would bring you closer. (Slightly) more rigorously, if you know how to derive the method of lagrangian multipliers, this is a similar idea.  so you can pick any line between these parallel lines to satisfy the problem.

If one is contained inside the other the above won't work, because even when a point of closest approach does exist, the parabolas will both be on the same side of the tangent line. It's a simple case though, just use the directrix of the larger parabola as your answer.
Here's a simple approach, which I hope will suit. 

Here is a parabola, courtesy of WolframAlpha:




Here is another parabola. 


Do the parabolas intersect? (No.) Is there a line, say y = -0.25, that won't intersect either parabola? (Yes, and there are others.)