No. Individuals do not issue form1099-Misc, only businesses.
They report the income from their own records. The IRS usually considers undocumented cash income (no W-2 or 1099-MISC), for work performed, to be self employment income. Enter at "Business Income & Expenses" and TurboTax (TT) will complete Schedule C for you and allow you to deduct any expenses associated with this income. You'll also have to pay self employment tax (social security & medicare tax) on any profits greater than $400.
BUT, as previously stated, parents are an exception to the nanny tax. They can report the income as Household employee income (instead of self employment) and avoid the self employment tax. But, they will not get social security credit for the earnings. So, they have the option of either reporting it as self employment or household employee wages. For how to do that, see https://ttlc.intuit.com/community/tax-credits-deductions/discussion/i-m-retired-and-on-ss-i-took-car...
Alternatively, you could file Schedule H. See https://turbotax.intuit.com/tax-tips/irs-tax-forms/what-is-schedule-h-household-employment-taxes/L7X...
