You can't take a tax deduction for expenses you pay to care for family members, regardless of their status or situation. At most, you can
1. claim a family member as a dependent, which gets you a $500 credit
2. deduct medical expenses you pay for them on your tax return, but only if you can claim them as a dependent.
To claim a parent as a dependent, they must be a US resident, and you must provide more than half their total support for the year, and they must have less than $4400 of taxable income.
Being a "US resident" does not require permission or legal status, but it does require the person pass the substantial presence test.
https://www.irs.gov/individuals/international-taxpayers/substantial-presence-test
Taxable income means income that is taxable to the US government. If someone is a US resident because they pass the substantial presence test, they owe US income tax on all their world-wide income. This includes business income, pensions, or "social security" type plans from foreign countries. (If someone becomes a US resident in the middle of the year, they owe US income tax on their world-wide income starting on the date they become a US resident.) Failing to pay US tax, if owed, can impact a person's eligibility for legal immigration.
If your parents meet all three tests, you can claim them as dependents. They would each need an ITIN.
https://www.irs.gov/individuals/individual-taxpayer-identification-number
You get the ITIN by filing your tax return by mail and including a form W-7 application for ITIN for each person. You mail the tax return, the W-7, and the required proof of identity all to the IRS. After the IRS issues the ITIN they will process the tax return.
