It's more natural to say "(they) stayed home to ..."
However, as Michael Harvey points in his comment, the funny bit is the phrasing of the last part of the sentence which invites an unwanted parallel between "cook" and "look after". For example:
I stayed home from practice today to clean and polish my new car.
Here both actions are implicitly being done to the car. In the same way:
... to cook and look after their children
can be humorously misinterpreted as "to cook their children" and "to look after their children".
Of course, no one will think that is what you meant, but if you want to avoid any misunderstanding, simply include the preposition:
Mothers, traditionally, stayed at home to cook for and look after their children.
(Side note) Peter's suggestion to substitute "care for" in place of "look after" is fine, but in my opinion it doesn't really clarify the meaning. The combination of mothers with look after is common in English writing.