...or would that childs' income need to be reported by the parents?
Income is not a factor, thus not reported for anyone in the family. Just household size and SS#s and ages and where and to whom to send the check.
Neither income nor wealth are taxed nor the basis of qualifying for the FCA.
As a point of fact registering for the FCA is not mandatory. If you don't want the FCA, you are not required to file squat.
Hmmmm, would Barbra Striesand apply for an FCA or not?