i am sorry to have to say this about anyone, but Baruch, how old are your cousins? They don't seem very bright to me unless they are 7 or 8yo then maybe they are just about normal. Otherwise, maturity level is very very low and, even though you are a young man yourself, you need to be the bigger man and completely ignore them. I mean it, these are pretty dumb jokes by any standard.
If you fail to convince your dad to help move the birds inside, would you be able to trolley the new cage in and out of the garage?
You need to do the best with what you got.
Good point 1 = you live in a warm climate.
Good point 2 = you can use the garage.
Good point 3 = you have an aviary.
If you get one of those great big cages that can be trolleyed around, you could keep the birds nice and safe in the garage at night and whenever you feel you can't be there to protect them.
The rest of the time, you trolley the cage outside and place them in whatever good outside space is near.
Also, you can still use the aviary to let them have a good fly around.
I don't know if that's doable, not sure about the floor plant of your house of course, but to be honest if I lived in Florida, I had an empty garage and an outside aviary that's how I would probably keep mines.
I understand that you want them inside the house with you, but I can also appreciate your parents' doubt about such a big group moving in.
If you manage with the garage and the big nice cage, you might be able to place a small cage in your room and ask for permission to bring one or two of them inside for a few hours; in a few weeks your parents would be more used to see them inside and see by themselves that they are not such a big disruption.
From there to a complete move-in, if you play it well I can see success in the future.
Of course I hope your talk with your father will have a positive outcome; but if it doesn't... well, consider different strategies. Don't give up
You could ask for exclusive use of the garage, after all it's used as a utility room, you say; if you can do that, you'd be able to transform it in a seriously great bird room.