A child arrangements order can show that a parent has direct contact, shared care or an active role in a child's upbringing. Those facts may be central to an immigration application. The same document may also be protected by the confidentiality rules governing family proceedings.
The potential conflict is practical as well as legal. An immigration adviser may need evidence urgently, while the family solicitor is managing proceedings in which disclosure to an outside body is restricted. The answer is coordinated permission and carefully defined use, not an assumption that immigration relevance overrides the court's control.