At first glance, it looks like Household Employees are not required to provide Worker's Compensation, but you may be required to pay State Unemployment and provide pay stubs.
https://www.care.com/homepay/resource-center/requirements-by-state/north-carolina
https://www.poppinspayroll.com/resources/northcarolina
Whether or not there are exceptions for a parent, I don't know. You really should contact a professional in your area regarding this.
