A child may be able to join one parent in the UK where the applicable Immigration Rules permit it and that parent has sole responsibility for the child's upbringing. The phrase is often misunderstood. It is not the same as sole legal custody, paying all expenses or obtaining a letter in which the other parent consents to the application.
The current Appendix Children caseworker guidance describes sole parental responsibility as sole control in setting and providing the day-to-day direction and care for the child's welfare. It directs the decision maker to examine who has made the important decisions about education, health, religion, residence, holidays, protection and guidance. Where both parents remain involved, the guidance says it will be rare for one parent to establish sole parental responsibility.
The application must therefore begin with the child's actual route and an honest account of how the family has functioned over time. If responsibility is genuinely shared, the correct question may be whether there are serious and compelling reasons for the child to join the parent in the UK. That is a distinct exception with its own evidential burden; it should not be obscured by an overstated claim of sole responsibility.