You have to click on the image of your pet, then a little box will light up, you can click and drag where you want your pet's headshot. That window (that looks like an item hovering info) doesn't affect the process. But it's nice to let know about it.
Originally when I clicked on the image that item hover window popped up instead of the headshot box. Nothing else would happen no matter how much I clicked. When I clicked and attempted to drag, a ghost image of the whole pet would move. I cleared my cache and refreshed, then switched from Chrome to Firefox and got the headshot to change, but that weird popup is still showing up.