More information is neede to answer your question. For example, how much monthly income does your parent have; how much long term care insurance coverage; what type of coverage; is there a living spouse? You should seek advice from an experienced elder law attorney immediately
I agree that it is important to meet with an elder law attorney in New York who can look at your parent's situation and discuss some planning options with you. Having long term care insurance will not necessarily disqualify a person from receiving Medicaid assistance. It depends on the amount of long term insurance received and the person's income. Another factor would be the maximum amounts of benefits the insurance policy provides.
Often the family has to decide whether taxes or preserving the estate from nursing home costs are are more important. You may be able to have one, but not the other. The critical facts that need to be disclosed to an elder law attorny are how much fixed income does your parent have and how much and for how long does the long term care policy pay.