Simplest answer: keep it in the family. They don't report income and you don't claim the credit.
Longer answer: if you want to claim the tax credit, you will have to give the IRS the social security number of the care provider or providers. The IRS will look for matching income on the provider's return. If the grandparents are retired on social security, then earning income from work might affect their social security, or it might not be taxable at all, depending on their other sources of income. It's really too complicated to cover all the "what ifs" in one short answer.
Your responsibilities
If they provide care in your home at your direction, they are household employees and you must issue a W-2 if you pay more than $2300 in a year. Because they are your parents, you don't have to withhold or pay household employee tax (social security and medicare) on their earnings. So presumably, if you paid each grandparent $2299, you don't have to do anything. If you pay more than $2300, you should issue a W-2, there are web sites that will e-file a W-2 and W-3 for you for about $5 each.
If they provide care in their home, they are independent contractors/self-employed. You don't issue any paperwork, no 1099 and no W-2. They are responsible for reporting their own income and expenses on a schedule C.
Their responsibilities
If you provide a W-2, they report the income normally. If they are household employees but you don't provide a W-2, they still report the income by adding it to their other income and writing "HHS" on the form. (It might be a different abbreviation, I can never remember the right one and I couldn't find it in 10 seconds of searching online, but Turbotax has a place to enter household employee wages not on a W-2 and will report it properly for them). If they are self-employed, they report the income on schedule C. They can deduct expenses, such as the cost of meals and a portion of their household expenses like utilities and insurance. They pay income tax and self-employment tax on the net profit. Having taxable income earned from working (earned income) after retirement has a variety of consequences, that are both potentially good and potentially bad, depending on their tax situation.
