Can I charge a very low rate to attract parents and not report it as income?

No way. Any money you receive from parents is taxable income. A few family child care providers try to avoid paying taxes by not reporting their income. They get parents to agree not to claim the child care tax credit in exchange for charging them a very low rate. Trying to hide from the IRS won't work. If you try this you are setting yourself up for penalties and interest when you get caught.