In the TV series Growing Pains, season 1, episode 6, there's the following conversation:

Mom: Mike, don't forget that your father and I are going out... with the Koosmans tonight and you have to baby-sit Ben and Carol.

Mike: Can't we get a sitter?

Father: On this short notice, not one without an axe.

Mike: So? Ben and Carol are quick on their feet.

What's the meaning of "not one without an axe" here? What does this have to do with an axe?