In which state do we apply for Medicaid when the selected facility is out-of-state?

If your parents own real property in the nonNH state, you may run into problems with their being OK for Medicaid in the NH state. Their house & car would probably be viewed as an non-exempt asset & as such would place their assets above the 2K allowed as the maximum or cap for assets under Medicaid.

Medicaid although a joint federal & state program, is administered by each state uniquely. The states seem to take the position that you have to be a true resident of the state to qualify - having a homestead in another state disqualifies you. They will need to have a drivers license or ID for NH state, valid mailing address, the documents required by Medicaid (bank statements, awards letters) will need to show NH state address, etc. I would really caution you to make sure that they will qualify before they move in otherwise they could face a huge private pay bill.

The state where the facility is.