Caregiver: someone who looks over and cares for a pet while regressed. You may also find caregivers referred to as handlers, which means the same thing.
Petspace: the term for when a pet is in the headspace of their animal, for example a bear regressor would call it bearspace.
Cg: short for caregiver.
Petre: short for pet regression.
Pet: a vague way of referring to a pet regressor.
Flip: someone who is both a pet regressor and a caregiver, they may be referred to as a switch.
Pet sitting/sitter: someone who looks after a pet when their caregiver can't.
Paci: slang for a pacifier, common gear for pet regressors.
Gear: the supplies used to help with regression and help pets feel more comfortable.
Slipping: when a regressor slips into petspace due to being in a less than convenient or extremely stressful situation. Of course this may depend on the context and this term could also be used in situations where the pet casually needs to communicate that they're going into petspace to their caregiver by saying "I'm slipping."
Sippy: short for sippy cup