OK, I think I see the confusion. The "nanny tax" refers to the requirement for the employer (you) to pay social security and medicare tax on her earnings. Being your parent makes you exempt from either of you having to pay any social security and medicare tax.
Income tax is a separate issue. There is no exemption for income tax being due on her earnings. So the wages you pay her are reportable, by her. However, a non-dependent person is not required to file a federal income tax return if they have less than $10,400 of total income (not counting social security benefits). So, since she has no other income, she will not be taxed on the $6000, nor will she even be required to file a tax return.