You are correct - a parabola requires the cutting be parallel to the edge of the cone. If the cutting plane is not equal to the slope of the edge of the cone then it will either be (a) too shallow and thus slice completely through the one lobe forming an elipse (a shape of finite size), or (b) too steep and thus its upper portion is always within the confines of the cone no matter how far you go, and its lower portion slices into tje lower lobe, thereby making two shapess that are each infinite in size. But if sliced at precisley the correct angle (parallel to the edge ) you end up with one shape that is infinite in size. Thus a parabola is the intermediate between an elipse and a hyperbola.

The wikipedia article is pretty complete: Conic section - Wikipedia, the free encyclopedia