Chikdcare

You are talking about having your mom work for you as a household employee.  If you pay someone to work in your home and they earn at least $2700 during the tax year, they are a household employee.   There are exceptions to the requirement to withhold federal and or state tax, Social Security and Medicare from a household employee's wages if the employee is your parent,  but you are supposed to issue a W-2 to a household employee.

https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-payments/pay-taxes-pay-household-employee/L9eZxdOGo_US_en_US?uid=m4mvnflf

Your mother's age is irrelevant, but when she and her spouse file their joint return, all of their combined income is required to be entered on their tax return, including the W-2 she gets from you.  So, just like with any other income, their joint tax liability is based on the total amount of income received and is reconciled with the amount of tax withheld to determine their refund or tax due.

Assuming you want to use the childcare credit on your own tax return, you must report the amount you paid to the childcare provider and enter that childcare provider's SSN on your own tax return.  The amount that you enter must match the amount the childcare provider enters on their return.

If you file a tax return for any reason, you are required to report all your income, no matter how minor some items might be.  If you pay her to work, regardless of where or how much, she is required to report it on her tax return.  How much tax she would have to pay depends on all her other tax factors.  

If she works in your home and you pay her more than $2700, you are required to issue her a W-2.  You are not required to withhold or pay household employee tax when the employee is your parent (in most cases, but there is a rare exception).  Even if you pay less than $2700 and don't issue a W-2, she is required to report the income, although it would be on the honor system if there is no report to the IRS that they can look over.