Here is a way to procede: what can be said about the intersection of a parabola and a line? You can easily prove that any line that is not parallel to the axis has a bounded intersection (a line segment, a point or the empty set) with the parabola and its interior. If you're given a finite number of parabolas, you can choose a line that is parallel to none of the axes, hence it has only a bounded intersection with all the parabolas and their interiors.