@Abvieon

When one of your currently placed pets is finished resting, you can go into your Pixpet inventory and there should be a "Remove Pixpet" button next to that pet. If your pet's status says "Resting: Finished!" but the Remove Pixpet button is unclickable, then you need to go to their hoard page (by clicking "See Hoard Details" or the pet's name or overworld sprite) which will manually end their resting phase.

Let me know if you have any other questions, or if anything I said was confusing