Does it mean the parents will allow them to eat anything they please? Absolutely not.

absolutely 是副词，意思是"完全地、绝对地"，用来加强语气。 not 是副词，表示否定，意思是"不"。 absolutely not 是一个固定搭配，表示强烈的否定，意思是"绝对不行"或"当然不"。