Two parallel lines never cross or intersect with each other and the (minimum) distance between them remains the same to infinity.
(Please note that I emphasize the “and” above solely to address the “binarity” of parallelism, for although “parallel” may not be binary in the sense that an infinite number of lines can be drawn that never intersect with each other, the constant equidistance part of the description of parallel lines can only be satisfied by a maximum of four parallel lines in a three dimensional Cartesian coordinate system and in the grand scheme of things, 4 seems closer to being binary than to being infinite.)
(link to a question on ‘Quora’ about the maximum number of points (and therefore parallel lines?) that can be equidistant from each other and one of its answers that says that the maximum is 4)
Contrary to parallel lines are the ones that eventually cross or intersect with each other and a tripod is formed when three non-parallel lines intersect/cross/come into contact at the same point, therefore I think “in intersection” (in the sense of “coming into/having contact with”) would convey the opposite of “in parallel” in the literal sense, but in a figurative sense, I would equate “in parallel” to “[standing] safely/comfortably apart” and its opposite (in tripod) to “[standing] dangerously close or too close for comfort”
For the sake of world peace, two opposing forces (world powers, for example) are perhaps best kept “safely/comfortably apart” from each other, running/standing [in] parallel with the [equi]distance between them maintained by, among other things, the notion of “balance of power.”
(To the extent that “being/working in parallel” might [or even has to] imply to some that the two forces have the same interests and goal, one could argue that they both have the same interests, i.e., self interest/preservation and the same goal, i.e., peaceful coexistence.)
On the other hand, the relationship between entities (such as the EU powers mentioned in the question) whose paths purposely cross and intersect on a regular basis, although giving rise to great opportunities to resolve Europe’s problems and achieve its unity, also gives rise to more opportunities for head-on confrontations and such a close relationship could be characterized as one that’s “dangerously close” or “too close for comfort,” perhaps similar to the feeling some participants experience in a three-legged race, especially when a virtual stranger has been assigned to be their partner.