Moving can be stressful for everyone, especially with animals!
With regards to your dog, it can be dangerous to put pets under the plane as cargo. Cages may be jostled around, and the temperature and conditions are often uncontrollable and unsafe.
Some airlines allow smaller dogs to be taken on board as long as they are in a carrying crate. If you have a larger dog, you may have to contact airlines to see which ones specifically cater to pets.
As for your bird, it is possible to take him with you. You would have to take him on the plane with you unless you absolutely trusted the airline, and they would consider him a "carry on" like a small dog, if they allow pets. It may be hard to find an airplane to where you are going that allows this, but that's the best option.
Which country are you moving to? Different countries have different quarantine rules so it's important to research those beforehand, since you will have to submit both your dog and budgie to a quarantine procedure before being allowed to enter the country (because of possible foreign disease spreading).
That's the best I can do
I'm sorry that your parents don't want you to bring your budgie. If you do find an airline that can take your budgie either in the cabin or safely
on board in another way, talk to your parents and see if it's something that they can do.
If your parents absolutely don't want to take the budgie, there's not much you can do if they're convinced of their decision. Since you will have to rehome him, I would begin writing down pre-screening questions you would ask any potential owners and consider that option, too.