Since you have already been allowed by the parents to handle them, it is OK to take the chicks to another room. I’m glad the daddy is so tolerant – sometimes the males can be more possessive than the females, in some cases even preventing the female from entering the nest box after the eggs have been laid! The abandonment issue comes up when there are major changes, such as relocating the entire cage or changing their habitat.
If you are asking about allowing the parents to fly free, with no outdoor enclosure, then no, this is not a good idea. I thought you were describing an area that is enclosed with screening or wire of some type. When Cockatiels escape from their homes, they are generally gone for good. They will fly for miles and miles and have no idea of how to get back home. Chicks will not be enough to keep them around. Cockatiels and other pet or captive birds will usually panic when they get loose and just fly until they land on something, fly into something or are brought down by something like a predator or a person with a gun. So if you set the pair free, not only would you probably never see them again, they would not remain together because they would be too scared to know what they are doing or where they are flying. These birds are presumably captive bred and do not have the instincts to be free. They will be caught by something eventually – a kind human if they are lucky, but more often something kills them or they die from starvation or disease. Most captive bred cockatiels are mutations now, or carry the genes which makes them less hardy than the normal grey wild cockatiel. Even if released in their natural environment they would not be likely to survive. Some people have trained larger parrots to fly free and return, but that often results in tragedy again due to animal predators or cruel humans who shoot them. The better option is to construct a large, safe enclosure outdoors if this is what you want to do.
Enjoy your sweet chicks!