To answer “If I get a bigger parrot like a cockatoo” is a fool’s errand.
An Umbrella cockatoo is roughly 18 inches long and weighs 1.4 lbs.
If I get a bigger parrot, like a cockatoo, should I be worried about it bonding with my budgies over me?
So if you are thinking about “a cockatoo”, I will advocate that you spend time learning about the 30 or so species of cockatoos with the Goffin or Tanambar being the smallest cockatoos roughly 12 inches long and weighing 9.7 ounces or half the size of an Umbrella cockatoo.
In spite of all the great answers, crystal balls don’t exist and no one can answer this.
If you clicker train all your birds which includes flight training chances are you may be successful with your endeavor.
You also may do better with a single large walk-in aviary vs individual cages.
The bonding process can take months or even years.
Start asking the “right” questions and your chances for success will increase.
BTW – cockatoos and budgies are both from Australia which doesn’t hurt.