You can claim the earnings on your taxes and the parents can claim the payments toward you so long as you are not eligible to be claimed as a dependent on their return.
You will file a Schedule C, as a self-employed individual. You can claim certain expenses, depending on whether or not you are a certified/licensed child care provider. If you are certified, you may claim the use of your home. You will be obligated to pay additional taxes (self-employment tax) if the parents paid you more than $400 for the year. The self-employment tax is approximately 15% of your child care income, AFTER deductions.
In order for the children's parents to claim the deduction for child care expenses, they will need an identifying number for you. You can provide them with your SSN or if that is not something you're interested in, you can apply for an EIN (employer identification number). You do not need one, however, it can be used to identify you instead of giving someone your SSN. You can read about applying for an EIN here: http://www.irs.gov/businesses/small/article/0,,id=98350,00.html
The parents will claim the payments to you on Form 2441, Child and Dependent Care Expenses. They will receive a credit. However, unless you paid estimated taxes for the year, you are going to be responsible for paying the federal taxes and self-employment taxes on this income.
Please let me know if I can help you further.