Certainly you should be treated as an employee of this couple.
http://www.usatoday.com/story/money/personalfinance/2014/04/12/taxes-irs-nanny/7123399/
I'd expect that they also will not issue you a 1099-MISC, either.
You'll report your income on Schedule C.
https://www.irs.gov/pub/irs-pdf/f1040sc.pdf
You'll use your own records to report the appropriate amount on line 1 ("Gross receipts or sales"). Any expenses you incur providing this service - minimal, I'd think - will also go on Schedule C. The "bottom" line of Schedule C on line 31 ("Net profit or (loss") will then flow to line 12 of Form 1040 ("Business income or (loss). Attach Schedule C or C-E").
In addition to income taxes you will also be charged Self Employment Tax (Social Security and Medicare) at roughly a 15% rate. However half of this tax will serve as a deduction on line 27 ("Deductible part of self-employment tax") in determining your actual taxable income.
You will need to use TurboTax Home and Business to prepare your 2016 income tax return if you use the online product as that's the only version that supports Schedule C. All desktop versions include Schedule C.
Because you will be receiving income against which there will be no tax withholding you might have to send in Estimated Taxes to the Treasury Department. (The US income tax system is a "pay as you go" system and most people "pay as they go" by having taxes withheld from their paychecks. People with significant income from which there is no withholding generally have to write checks and submit estimated tax payments to the Treasury.) Estimated taxes are due more or less quarterly generally on 4/15, 6/15, 9/15 and 1/15 of the following year. (The 4/15 date is not a problem for you since you've not yet earned any income from which taxes haven't been withheld.)
Assuming you used TurboTax for your 2015 income tax return you can use that program to see if you need to pay estimated taxes in 2016:
https://ttlc.intuit.com/questions/1900956-can-turbotax-calculate-the-estimated-payments-for-next-yea...
Tom Young
