You are correct. A 1099-MISC is how to report that income.
Here's a link to fill those out:
Quick Employer Forms
The caretakers are responsible for their own federal income tax and all (15.3%) of their social security tax.
The elderly taxpayer can deduct wages paid for nursing care (see Nursing services, in this:)
IRS-Approved
Medical Expenses
You may want to go down the list in blue on the left of the publication: there may be more expenses that you overlooked.
