Desktop: Login to your account and Click on the My Pets heading on the left had side menu to see all your listed pets. Select Options under the picture of the pet you wish to remove.
A small pop up will appear. Select the REMOVE PET option.
Another pop up will appear that asks you to select the reason for removing this pet. Choose the most appropriate option and then click on the purple REMOVE button at the bottom and this will remove the pet from your account.
Mobile: Same as Desktop instructions.