This feature is currently turned off. The inability to Edit Pet Profiles is currently an Integration setting done in your respected Shelter software, which always overwrites information from the shelter software to Adopets to keep everything up to date. You need to update the information through the integration system that was provided. Once done, it will sync in with Adopets.
You would not be able to edit the pet information in Adopets. Because of this, it would also stop the update of data from the pet in the Shelter Software to Adopets, making the integration of the pulled data from your pets only once, when they are imported to Adopets.