The fact that you have two of each type bird is very positive as they will at least have one another -- whichever way you choose to handle the move. The fact they are not budgies is also a positive as both species are larger and not as fragile.
If any of the birds in question was a single bird that was closely bonded with your (or your parents) that would be a much more difficult scenario.
Parrots can and do have negative reactions to separation from their owners. The more closely bonded the bird is to the owner, the more difficulty it will have in dealing with a separation. Sometimes the reactions manifest as behavioral issues such as overpreening, self-mutilation, aggressiveness or depression.
If your parrots are used to being left with other people for periods of time, such as when you go on vacation for two or more weeks, that be advantageous in the upcoming situation.
In such a case, I would not be as concerned about them being able to handle the change of routine and enviornment as I would otherwise.
If, however, they have never been away from you and your parents for any length of time, then the separation (whether it is for one month or for five) will be more difficult for them. In this case, it is going to be extremely important to take into consideration the personality of each bird and the potential effect the separation will have.
It is crucial for you and your parents to sit down and carefully weigh out all the pros and cons for both potential scenarios, evaluating the birds' personalities, the risks involved and the potential advantages of each of the two choices.
If you transport the birds by cargo plane, temperature, time in the air and how the birds will be handled during transport is a concern. During the quarantine period, where will the birds be kept, who will be caring for them, what kind of human interaction will they receive, will they receive their regular diet with a normal routine and will you be able to visit them? Can your family afford the fees necessary for this scenario to be a viable option?
If you leave the birds with the vet you know in Mexico, will they be housed at the individual's home or at a pet clinic? Will they be given attention as if they are part of the individual's family of pets or will they simply have their most basic needs of food, water and cleanliness met for the five month period? Will they be exposed to other birds during the time period? Won't they still have to be quarantined for a month when they come to the U.S. at the end of that time?
These are obviously not questions I expect you to give the answers to on the forum, but are serious matters you and your parents will need to consider as you weigh the decision before you.
I wish there was an easy answer but there are too many variables that I have no knowledge of for me to give you a direct opinion on how to best handle the situation.
Sending lots of positive thoughts to you and prayers for you to have the strength and wisdom to make the best choice for your four parrots. :hug: