Yes. Any change to a written contract must be in writing for it to be enforced. So the parent can't enforce your verbal agreement because it conflicts with your written contract. If the parent has not paid you for the three weeks, she may not be happy about paying now since you went back on your word. Enforcing your contract in court would not be a sure thing, even though you are legally in the right.
Before you make a generous offer to allow the parent to leave at any time, make sure you are fine with the potential loss of money if the parent decides to leave immediately. In the future you should require the parent to pay for the last three weeks in advance. Then you are in the stronger position to refund some of the money or not.