The family said they were tackling the dispute with the council while supporting eldest son Oskar, 17, while he received treatment for blood cancer.
Karen Olejarka said they pleaded guilty as fighting the case would have been too expensive.
“We’ve gone from fighting for our second son to be able to live his life, to fighting for our oldest son to live,” she said.
At an earlier hearing, Hampshire County Council said it had attempted to resolve the issue before prosecution, but prosecutor Ian Jenkins said Dexter’s father had “gone nuclear” and made it a national issue.
Child employment in England is covered by the Children and Young Person Act 1933, and also by local byelaws drawn up by each local authority.
While Hampshire’s byelaws state that “children may do light work from the age of 13 years until they are no longer of compulsory school age”, employment in commercial kitchens is prohibited until the age of 16.
The council said: “We will only consider legal action in the most exceptional circumstances, where there are safeguarding concerns that cannot be addressed in other ways.
“The defendants in this case pleaded guilty and they appear to have recognised that the employment of Dexter in a commercial kitchen was unlawful.”
Dexter’s parents said he was no longer helping out on the food trailer.