I must respectfully disagree with degreecy. Whether or not children are revenue or mileage-paying passengers is irrelevant to the issue at hand. Essentially that argument leads in the direction of (bluntly put): “I paid for it, so I can behave however in the hell I want.” Despite some people unfortunately believing this, most people will agree that money doesn’t give one the right to break the rule … whether those rules are written rule or social normal of common courtesy. Disregarding those social normals is either a blatant attack on those around the offender or a lack of self awareness.
I believe there is a time and a place for almost everything. I, an adult, shouldn’t show up at a children’s playground to read a book and then complain that the children are running around making noise. Likewise, parents and children shouldn’t come into a business or first class cabin — utilized mostly be people that value on-board privacy, calm, ability to work or relax — and utilize the space as they would their playground.
I’m sympathetic that sometimes outbursts are unexpected and uncontrollable … sometime my dog jumps on people and I wish he wouldn’t. Sometimes babies cry on airplane despite their parents best efforts to comfort them. In both cases the responsible adult must take appropriate action to ensure those in their charge are not disruptive to innocent people around them.