I never put water in their carriers. My birds drink water once in the early morning and again once in the pm. They never drink in the middle of the day but, then, they eat a fresh food diet so they get plenty of moisture from it and birds that eat pellets need much more water so they might drink in the middle of the day (I don't know because I don't ever feed pellets so I have no occasion to observe a pellet-eating bird for any length of time and compare it to mine). What I do is put orange slices and celery sticks as well as some millet in the carrier. The orange has a lot of water, the celery is high in sodium so they retain water and the millet is for comfort and entertainment.
But don't worry too much about the trip, parrots are much more resilient than people give them credit for. I have travelled 9 hours in a car with a parrot in a cage on the back seat and have flown from California to New Jersey with another one which pretty much took the entire day because I made a long stopover on purpose so I could let her out of the cage for a couple of hours.
Cover the carrier or cage you will be using to get it out of the house and into the car. Use the seat belt to hold and steady the carrier or cage and uncover just the front so the bird can look out but still feel safe in his 'little cave'. Talk or sing along with the radio (my birds are all so used to my singing a couple of songs that they have become like the crickets in the forest, as long as they hear them, they know everything is OK) and stop halfway if for nothing else to let him out for a few minutes to cuddle with you (bring his favorite nut or whatever with you and give him a few then) but be careful not to open the car door!.
The vet will know what he needs done but, FYI, a complete physical (I don't know when was the last time if ever he had one) means physical checkup by the vet (sometimes they even take their blood pressure), CBC (Complete Blood Count), avian chem (Avian Chemistry panel) -sometimes they offer what they call a Complete Avian Profile which is the two panels together for one price, Bile Acids test (the only test that measures liver function) and full body Xrays (usually two, one from the side and one from the top).
Good luck and let us know what he says.





