No, none of the psittacines (including cockatoos) have gallbladders. Nor do pigeons, or ostriches.
They have two hepatic ducts (a right and a left, coming from the respective two lobes of the liver). They both empty directly into the duodenum (the first part of the small intestine).
Occasionally, the right hepatic duct is found to be slightly dilated (especially in cockatoos) and thus has the appearance of a rudimentary gallbladder, but it is not one, it is simply a dilated hepatic duct. There is a microscopic difference in the arrangement of the epithelial cells lining a hepatic duct compared to those lining a gallbladder - this is how this fact is known.
The cases of mildly dilated hepatic ducts are not pathological, but are simply considered to be an anatomic variation.