Medicare coverage is federal but providers are state

You are referring to MEDICARE not MEDICAID, correct?
Your parents are covered under Medicare in Florida but here is the "glitch" because the same thing happens to me.
Each state "farms out" a contract to individual providers. All you parents need to do is record the name and address of the NYC provider that is attached to their Explanation of benefits, EOB's.
Give them a call ahead of time to save you the grief of having to do this several times and know the procedure before you need to utilize it.
If they need services in a different state they need to provide THIS address at the time of service. Sometimes it doesn't always work, limitations of medical personnel, but you still can fix it.
Wait until you receive the bill and forward it personally to their medicare provider in NYC.
I'm from Florida, I ended up in the hospital in PA, it took six months to clear this up but it turned out that all I needed to do was to give the hospital the correct address for the Florida provider. After that they took care of everything.