The real answer is "It depends..."
There are many more factors that need to be considered than what has been discussed.
As others have mentioned, the general rule is that officers can question a child, or any other person, about anything, anytime, anywhere. Whether any statements can be used against the child or person questioned, or must be suppressed, is where the discussion thread has gone so far. The answer is that it depends whether the child was the subject of the investigation, whether it was a child welfare investigation, whether the questioning was coercive, and all the circumstances discussed in this thread factor into determining whether any answers will be admissible.
Generally, if the child is the subject of a criminal investigation, the parent can demand to be present, and courts scrutinize the circumstances surrounding the questioning to determine admissibility of any statements the child made.
if the child is not the subject of the investigation, the person who IS the subject of questioning does not have legal standing to seek suppression, because his rights have not been compromised - it was the child who was questioned, not the subject of the investigation.
It really gets complicated when the police are conducting a child welfare investigation - child abuse, child sexual abuse and the like. Usually in such investigations, it is a parent who is the subject of the investigation, and most states hold that the child's welfare trumps any demand by a parent, particularly the one who may be abusing the child, who has insisted to school officials that they be contacted or be given the opportunity to be present during any such questioning. Allowing the abuser to be present to intimidate the child victim during questioning would further victimize the child, obstruct the investigation and lead to continued abuse. No state, to my knowledge, has ever held that a school official or school district must honor any parent's demand to be notified or present during any child welfare investigation, or allowed a lawsuit filed by a parent whose child was questioned against the parent's wishes in a child welfare investigation.
Different story, though, if the child is questioned about anybody or anything, where the child is not the subject of the investigation. For example, consider the case where the police interview kids about who is dealing drugs in the neighborhood, or who might be behind a string of burglaries or vandalism. In that case, I submit that schools have an obligation to contact the child's parent-regardless whether the parent has given school authorities notice they wish to be present during any questioning of their child - and not permit any questioning unless and until the parent is present or gives permission for the questioning.
Just so everyone knows, schools are the primary places where police and social workers make contact with children to interview them in child abuse or neglect cases. Parents can demand to be notified or present during questioning, and in most cases the demand will be honored, unless it is a child abuse or neglect case and the child is being viewed as a potential victim of abuse or neglect. In that case, logic and public policy dictate that the parent, who is the real object of the investigation, is not entitled to be present.