You need to give the money to the parents, and they will be responsible for paying the nanny, and withholding and filing household employee's tax.
If you directly hire this person, you are not a household employer, you are a regular employer (because the worker is not working in your home.). You would essentially be acting as an employment agent. You would need an employer identification number, and you would need to withhold state and federal income tax, social security and medicare tax, unemployment insurance, worker's compensation insurance, and who knows what else. There may be other state requirements like minimum wage, or required documentation and benefits. It's going to be way too complicated to go through.
If your child can't be trusted to handle money and do the paperwork correctly, it would be better for them to hire a nanny through an agency. The nanny is an employee of the agency, so the agency handles all the paperwork, and you can pay the agency directly with no paperwork requirements on your end.
