It is always better to have fully dried the perches prior to setting them back in the cage. That said, it sounds like you did a light wash and rinse of the surface and that the perches did not sit in water or worst, a cleaner for an extended period of time. So, setting them back into the cage and allowing them to finish surface drying will not cause harm to your Parrot. However, this should not become a standard practice! Standard Practice is to allow everything to fully dry.
Disclaimer: Yup, there is always a Disclaimer! Perches that have sat in water for an extended time period will take an extended time to dry and placing them back into use prior to fully drying is not recommended!
It is always better that surfaces that the Parrot will perch on are kept clean and dry. The best way to dry the cage, its accessories, including perches and toys is to allow them to dry in bright Sunlight NOTE: Wet surfaces grow 'stuff' rapidly!
I strongly recommend that you target branches that are cut live from a standing tree or a tree that has been recently fallen! Avoid branches that have laid on the ground after being either blown from or as a dead branch, fallen from the tree. Avoid any branch with mold on it! Always remove the bark, provide a quick wash and rinse and let dry in bright Sunlight. This will assure any 'bugs' that maybe on the branch are dead prior to bring them into the home.