Source your birds from as close to home as possible, to avoid long car journeys. Larger birds will be able to travel in cat or dog carriers (but only ones that have been thoroughly cleaned, if there was a previous mammal occupant). You can also buy purpose-made parrot carriers - these will be useful for taking your bird to the vet. They are usually either solid-framed, folding cages, or soft structures with supports, a bit like a tiny one-man tent. Smaller birds can be carried home in a cardboard carrier - the shop of breeder should be able to supply one of these.
New birds, like this Yellow-Headed Amazon, will need a carrying box or cage
Trips to the vet are an important part of parrot-care, so a travel cage of some kind is a piece of kit you can’t do without. Covering the cage during the journey is a good idea, and it will need securing with a seatbelt. Don’t put it where it might slide around (including the boot, which may also accumulate deadly car fumes). If it’s a snug fit, you might be able to fit a smaller carrier box in the glovebox.
If you haven’t had the bird for long, you may have to use the bird’s existing cage for trips in the car (although if the cage is big this is not a viable option). Remove anything that can move around - non-fixed bowls, swings and hanging toys - and make sure there’s a secure perch. The trip is going to be stressful for the bird, so minimising that stress is your priority.
If your parrots are in a larger aviary you will need to have acquired the skill of netting and re-caging 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.