Put the parakeet-carrying box somewhere safe and secure, where it won't move or fall. The lap of a car passenger is an ideal place, or the floor space beneath the passenger seat. If you are putting the box on a seat, use a seatbelt secure it.
Never put the parakeet in the trunk of a car. The exhaust fumes that accumulate there will kill it. If you can fit the bird-carrier snugly in the glove box, that’s a good option.
How to Transport a Parakeet to the Vet
You should use a custom-made small bird carrier cage for any trip away from home - and those trips should be limited to necessary ones such as trips to the vet. The carrier should a fixed perch and padded sides - you can buy ones made of foldaway nylon. It helps if the vet can see the parakeet before examining it in the hand, so carrier cages are better than cardboard boxes. .
A carrier cage suitable for parakeets
If your bird is still young and timid, or has never been hand-tamed, you may have to transport it in its existing cage (although if it's a large cage, this is not a viable option). One of the key requirements is a fixed perch, to help the bird feel secure. Use a seat belt to prevent the cage sliding around, and put a cover over the bars to help your pet feel less panicky.
If your birds are kept in an aviary or birdhouse, you will have to net and and cage them for the journey. This definitely needs an expert hand, and by far the best option would be for your local bird vet to pay you a visit.