If you are the parent who has actual physical custody of the child more than 183 nights of the year, then you are the only parent with the automatic right to claim a child as a dependent. The IRS defines custody based on where the child actually lives, and does not follow court orders or custody agreements.
If you are the parent with custody more than 183 nights of the year and you want to allow the other parent to claim the child, you give the other parent a signed form 8332. They must include the form with their tax return. It doesn't matter if the other parent doesn't have an SSN or an ITIN, just give them the form and leave it blank. Getting a number and filling it in is their problem. If Turbotax won't allow you to print a form 8332 without the SSN, you can get a blank form from the IRS web site. https://www.irs.gov/pub/irs-pdf/f8332.pdf
If the other parent needs a tax number and is not eligible for a SSN, they can apply for an ITIN (international tax ID number) using form W-7. They will fill out form W-7, attach any required proof of identity, and mail it to the IRS with their completed tax return, to the address on the W-7 instructions. The IRS will process the ITIN first, then add it to the tax return and process the tax return. The person won't be able to e-file.
Even if the person is in the US illegally, they are still required to file a tax return and pay taxes. Tax returns are confidential and the IRS will not inform on them to Immigration.
