When photographing people and their pets, have everyone lie down on their stomach including you, the photographer. It's a wonderful way to get personal with your subjects. I love portraits where faces are side by side and touching. It's a terrific way to show that special bond and connection people share with their pets.
Ask the person to lean into their pet, hug their pet, cradle their pet - whatever feels comfortable for them. Let the love flow and capture what they give you.
Keep it simple, relaxed, have fun and make it memorable.