Yes, it is allowed. But you probably can't e-file. In the space for the social security number of the care provider, you need to write “LAFCP” (Living Abroad Foreign Care Provider) and I don't think you can e-file when doing that. You may need to use a fake SSN, print your return, and use some white-out to cover the fake SSN and write in LAFCP. Then sign and mail your return.
Also, the foreign country (Greece) may have rules about paying someone to provide child care. For example, in the US, you don't have to pay the nanny tax if the employee is your parent, but you do have to issue them a W-2, and they have to report the income on their tax return (depending on their age and other tax situations). You will want to check to see if Greece has similar rules about paying a relative to work in your home.
You don't need to submit documents to the IRS when you file. If audited, you would need proof that you and your spouse were working, and required child care so you could work, and you would need proof that you paid your parent. That might include your pay stubs, information from your employer about the hours you worked (such as a written WFH agreement if you had one that specifies duties and expectations), copies of bank records showing either cash withdrawals or electronic transfers to your parent, or a copy of your parent's tax return showing that they reported the income.
