Maybe one must not think of light rays, a classical concept and hopefully parallel light rays are well derived mathematically as discussed in the link you give, but rather of photons.

Photons are elementary particles and light ( and its geometrical representation into light rays) emerges from the superposition of the complex wavefunction of each individual photon with the zillions that compose the classical electromagnetic wave. see here and links for an explanation.

If a photon is on the horizon, it will either keep on going down, or be trapped to be going around as in the photon sphere.If a photon is not going in "parallel" with the orbit of another photon, their paths will intersect, by definition of parallel. There is a photon photon interaction, and depending on its energy it can either create particles ( high energy) where they will be swallowed by the black hole or emerge as part of the hawking radiation, or might just scatter off each other elastically, changing directions, with very high probability of one going down into the black hole and the other leave and escape if it has enough energy, again being part of the hawking radiation. Thus over time, the only trapped photons on the horizon have to be in parallel orbits around the black hole.

If all photons are in parallel trajectories, the light ray will also have to be in parallel trajectories, as it is emergent from the zillions of photons.

This argument of course holds for the photon sphere too . This is a hand waving argument, if you do not want to go into the mathematics of the subject given in the link you have.

Maybe one must not think of light rays, a classical concept and hopefully parallel light rays are well derived mathematically as discussed in the link you give, but rather of photons.

Photons are elementary particles and light ( and its geometrical representation into light rays) emerges from the superposition of the complex wavefunction of each individual photon with the zillions that compose the classical electromagnetic wave. see here and links for an explanation.

If a photon is on the horizon, it will either keep on going down towards the singularity, or it will be trapped to be going around as in the photon sphere.

If a photon is not going in "parallel" with the orbit of another photon, their paths will intersect, by definition of parallel. There is a photon photon interaction, and depending on its energy it can either create particles ( high energy) where they will be swallowed by the black hole or emerge as part of the hawking radiation, or might just scatter off each other elastically, changing directions, with very high probability of one going down into the black hole and the other leave and escape if it has enough energy, again being part of the hawking radiation. Thus over time, the only trapped photons on the horizon have to be in parallel orbits around the black hole.

If all photons are in parallel trajectories, the light ray will also have to be in parallel trajectories, as it is emergent from the zillions of photons.

This argument of course holds for the photon sphere too . This is a hand waving argument, if you do not want to go into the mathematics of the subject given in the link you have.