Can planets have orbits other than elliptical or circular orbits?

Orbits are conic sections therefore can be either circular, elliptical, parabolic or hyperbolic.
Of these 4, only first two form a closed curve under 2 body hypothesis, while the later two extend to infinity.
If you talk about planet, by definition it has to orbit a star which would require it to have a closed orbit hence circular or elliptical, with respect to the star. For any other kind of orbit the body will just fly away to infinity never to return back. Infact comets are considered to be parabolic, but in theory they have highly elliptical orbits with their aphelion lying near the edge of the solar system.

But it is possible for a planet to have other kind of orbits if we consider their motion from a different reference frame such as with respect to another planet.
So for an inertial frame of reference a planet will have a circular or elliptical orbit, even Pluto.