Are children's rights to have their parents care for them a positive right or a negative right? My assertion is there are no legitimate positive rights. If our difference is on the definition of rights, and you are colloquially using the term, then yes I agree.
Discussion
It is a right in the sense that to withhold requisite care the parents would violate their child. Whose responsibility it is to be sure the parents do provide said care is a separate matter worth debating, but the basic point is that the parents ought to provide it.
To say the parents aught to care for their children is like saying a bear aught to care for its cubs. If we are to extend a positive right to the children to force their parents to do what they should naturally do, then we must use force to make their parents do what they apparently aren't doing. Its better, in my opinion, to think about it from the children's perspective. If the children do not want to be with their parents, and the parents do not want to be with their children, then the state (or society) should not force them together. If the parent is relinquishing thwir property rights over their children through neglect, then the children are free from the bondage of their parents by default. There is no need to provide rights to the children under the care of their parents in this case, because at that point the children have been emancipated defacto and they are free to find another means of survival such as from an orphanage. In the case of an infant or small child, like the second ammendment, its better to err on the side of freedom and property rights than on the side of the state "protecting" other people's property.