Long period comets seem to have orbits that are parabolas. The earth's orbit is almost a circle. But, a circle would be a perfect shape. Things like that are never perfect. The orbit of the earth differs from a circle a little bit, it is an ellipse.
Some ellipses are much flatter than the earth's orbit.
The orbit of a short period comet is a very flattened ellipse. A parabola is essentially an infinitely long ellipse. A parabola is between a very long ellipse and a hyperbola. A comet which comes from some distant place, well beyond the Solar System, should have an easily identifyable hyperbola for an orbit. The Solar System is moving through space at a fairly high speed.