Employees have legal rights to take time off work in certain circumstances involving their dependents. This currently applies to immediate family or other people living in the same household. Household pets should be recognised within this. For people who have pets, their pets are dependents.
You may be interested in these active petitions
Recognising pets as dependents would allow employees to take emergency time off for dependents for unavoidable appointments for household pets to the vets or vet hospital in emergency, without being penalised.