Modern human civilization pushes back and exploits nature quite a lot. This adversely affects not only the connection between humans and nature, but also the connection between animals and nature, as well as the link between humans and animals.
A pet is an animal which has limited freedom and relies on its human master for fulfilling its basic needs. Additionally, a psychic pet develops a symbiotic relationship for spiritual progress with its owner.
Even in older times it was common for a psychic or spiritually advanced individual (eg. witch, mage) to have a psychic pet. (The term 'familiar' was used instead of 'psychic pet'.)
A pet psychic can observe and develop a relationship with his or her psychic pet, and this process can serve as a gateway to understanding a different, more basic reality. Human social and intellectual norms are often a hindrance to psychic development - a psychic pet can help see human mistakes for what they are.