The property could have been sold for back taxes, or it could have been condemned for City purposes, or it could have been sold outright. It is possible that another City agency acquired the parcel and then it was later transferred to the Department of Parks.

I recommend that you contact the archivist of the City of New York Parks Dept., located at the Arsenal in Central Park - you can get particulars from their website. Their files should have information on the property and probably how it was acquired. I'm not sure how to research thru the Dept of Finance on old tax sales, and you have to do research in person at the City Register's Office in Queens where the land records are kept. At the Register you would locate the block & lot number and research that particular parcel, or you can search thru the Grantor indexes for your family's name, and then search for deeds on the microfilm, based on the info you find in the index.

Keep in mind that if the family owned real estate and even if there was a tax lien on it, there were obviously children entitled to the estate. Someone had to have been authorized to administer the estate even if there was no will - the Surrogate's Court would have issued letters of administration to someone to dispose of real property and any other assets.