Is there any way to get my parent's property taxes decreased or eliminated?

In NYS there is an enhanced star program based on income that reduces the school tax by about 30%. A tax return has to be provided early in the year and if approved takes place the following year. This year NYS residents received a tax rate rebate if their area had kept their property tax increases below 2%.
As your parents are qualified for medicaid their income will be low enough to qualify. They can of course grieve their taxes if they feel theirs are higher than similar properties in their area.

Most likely there is. They or their POA needs to go by the county tax office with the correct forms -- tax returns or disability letters from doctor, proof of age, POA forms. Property tax exemptions usually take effect the year after they are applied for. There are some exceptions to this, so check with the county's property tax office. Good luck.