If you balk at your mother's suggestion that you take on more responsibility, you're saying no to added chores. To balk means to refuse to go along with.
A donkey balks when it refuses to move forward. This is a good picture for balk which is often used in conjunction with demands. Demands are something people often balk at like a donkey refusing to move. In baseball, a pitcher balks when he or she begins a pitch by winding up, but does not complete it. It is as if he is refusing to complete a started pitch, and it is against the rules.