Here are all the permissions provided by MCPets and what it allows :
mcpets.use : enable GUI opening, pet interactions
mcpets.admin : access others' pet interaction menu, spawning pet to others and all other commands
mcpets.color : enable colors in pet's name
your.custom.pet.permission : each pet has its own access permission, so you can unlock them to players through quests, events, etc... You can find and change that permission into the pet config.