At present a lady bird deed should be sufficient to prevent the NYS Medicaid program from seeking to include the condo in your parents' estates for the purpose of recovering medicaid costs from their estates. I do not have extensive experience in this area, but my research indicates that in 2012 New York repealed the legislation that had allowed for an expanded definition of the term "estate" for the purpose of allowing the medicaid program to recover from medicaid recipients' estates. There is a short blog entry on this subject here: http://estatetaxes-ny.com/2012/03/27/no-expanded-estate-recovery/ (please note that I am not vouching for the accuracy or currency of this information; please check with an attorney you've retained for this purpose before relying on it).
As such, it would appear that New York's ability to recover medicaid costs from a recipient's estate is back to the former rules, under which the state medicaid program can only recover from assets that pass by will or intestacy.
Because a Lady Bird deed results in the deeded property passing outside of probate, it would seem to be the case that NYS would not be able to recover medicaid expenses for your parents from the condo.
That being said, it is always impossible to know all of the relevant facts in a forum such as this, and since the stakes are so high in your situation, you really, really should find a local estate planning or elder law attorney to speak with.