My parents are transferring long-term care facilities in Oklahoma, on Medicare/Medicaid. Are we allowed to take the current medications with?

You need to talk to the facility they are entering. Moms AL told me not to bring Moms meds they would get scripts from her doctors. Well, that did not work because records at the pharmacy showed Mom still had pills left and would not refill. Yes, I knew that would happen but the Nurse felt there would be no problem. So, I had to take her pills in.

Sure you are, but they will have to be reordered in a new facility by whatever MD is the treating physician. So it is crucial that medications are not just stopped while that occurs.
Check now with the receiving facility how it will all go with the medications that the persons are taking daily.
Best of luck to you.

My dad had a volunteer job for many years in which he picked up unopened, unused meds from area nursing homes and took them to a local charity health clinic for their patients who couldn’t afford needed meds. The nursing homes were glad to help and I wonder if they’d also be glad to send your parents along with their current supply of meds as well. There might need to be some method of assuring no improper taking of meds could happen. It’s worth asking the director of nursing where they currently live