Pet Parenting

Meaning: a person who looks after a pet animal.

The term pet parent is considered by some people concerned with the rights of animals to be more acceptable than owner. Pet parenting also refers to the responsibility and care that individuals provide to their pets, treating them as members of the family. This includes providing food, shelter, exercise, grooming, veterinary care, and love and attention.

Source: Collins Dictionary

https://www.collinsdictionary.com/dictionary/english/pet-parent
Further Reading:

1. https://www.petmd.com/dog/care/evr_dg_responsible_pet_parenting
2. https://www.thesprucepets.com/what-is-pet-parenting-4589032
3. https://www.thesprucepets.com/how-to-be-a-good-pet-parent-4767024