Parole has many meanings, including ability to speak, utterance, (spoken) word, and in plural, lyrics. It also shows up in expressions like "word of honor" (parole d'honneur).
As opposed to "mot" (word), not only is "parole" usually a bigger unit of language (speech, utterance), but there also seems to be an emphasis here on what is or has been spoken. Kind of like when someone says you have my word! Here then I think "parole" would be used as opposed to "mot".
I gathered from what I heard about De Saussure that "langue" was the word for language as a system, and "parole" was the word for the concrete uses of language; so I assume that here "the language of children" would be a correct translation as long as it was implicitly assumed that we were thinking about language in a concrete way; i.e, "what children say". Is this correct, or is it wrong, or am I just applying a too specialised meaning of the word "parole" to an everyday context?