You don't need an EIN at all. For a joint return, you are "one taxpayer" (more or less) so it doesn't matter which spouse the employee is the parent of.
Because the household employee is your (one of your) mother's, you are not required to withhold or pay household employee tax (social security and medicare tax). Because you do not withhold social security and medicare tax, you are also not required to issue a W-2 for her household employee wages. And that means you don't need the EIN.
(If you withheld federal or state income tax, then you would issue a W-2, even though boxes 3-6 would be blank, and then you would need the EIN. You would also need to file form 941 or 944 to report and pay the quarterly federal income tax withholding to the IRS. But this is a lot of work you don't need to do. It would be far easier to just pay your mother and let her make estimated tax payments, if required.)
As a household employee without a W-2, your mother would report the wages on line 1b of form 1040, "household employee wages not on a W-2." Turbotax includes this in the standard interview.
