I found this to be true. Good news is I never felt guilty about it. Only people who have had a pet they bonded with and then lost can really get this thing about grief. IMHO
Never having had kids I believe it is similar to losing your child.
Your pet is there with you every day and it is a deep symbiotic relationship ... they need you for food, shelter and in return give companionship and you share 'love' ... your parents and friends are around, but maybe seen a few hours a week, often much less.
I often felt guilty for feeling worse over my pets illnesses and death than people...