I think people who refer to themselves as a dog's mum or dad make me sick
Is there a reason people referring to themselves as these would upset you?
Dogs are animals, people are humans. What would you do if you were treated like that?
While animals and humans may not be the same, people can feel strongly about a pet and love it accordingly. You shouldn't judge others for  caring for their pets like a parent would for a child. If it makes them happy then it shouldn't matter to you.
I do care about animals, I'd just never put them over people.
Just because somebody refers to their animals as their kid, it doesn't mean that they value animals more than humans. Try to be more open minded and less judgmental when it comes to animal lovers.