Hi,
It sounds like the parents may be plucking the chicks. This is not uncommon. Is this their first clutch? Inexperienced pairs may do this, but also a pair that has been allowed to breed too often. Once these chicks leave the nest, the box should be removed and the parents rested for 6 months before you let them breed again. If the feathers do not grow in or if the wing and tail feathers do not grow in, the chicks may have a health issue. There are viruses that budgies can carry that can affect the feathers. Usually when the feathers do not grow from the start, it is Polyoma. A vet can test for these viruses, but there is no treatment. If the chicks have this, it comes from the parents and the pair should never be allowed to breed again and should be kept away from healthy birds.
Thank you for asking Lafeber,
Brenda