Conor ) gurgled happily, sounding much like they
were saying :

If not their child, what relationship *are* you to your parents?

Son, daughter?

and non-gender-specifically...?

If the supermarkets don't say what they mean on the signs, how are we
to know what they mean?

A "child" is a person under 18.

That's one definition of the word, but not the only one.

http://dictionary.reference.com/search?q=child
You might like to check up on definition 4.

Perhaps the supermarkets should clarify?