There are four possible types of degenerate conics (Wikipedia): a point, a single straight line, a pair of intersecting lines, and a pair of parallel lines.
I am however unable to see how one can use a 2D plane to slice a double cone to get a pair of parallel lines. Googling, I'm also unable to find any image showing how this can be done and I was hoping someone here could show me (if this is indeed possible).
I did find this decades-old discussion claiming that the "solution" is to treat a cylinder as a "degenerate cone". Is that the correct solution?