I relation to demandingness, Diana Baumrind says that it refers to "the claims parents make on children to become integrated into the family whole, by their maturity demands, supervision, disciplinary efforts and willingness to confront the child who disobeys". (Baumrind, 1991).
Another word for demandingness is control. The demandingness continuum (high vs. low) describes the level of behavior control parents exercise on their kids based on their expectations of 'mature' behavior.