It is frustrating beyond words that Brooklyn did not put parents' names on death certs until after it became part of greater NYC in 1898. Even then it took awhile because they thriftily used up the old forms, as I have found to my disappointment.
But it can sometimes be done with a lot of time and patience. Tho a death/funeral newspaper notice might not have the info, sometimes there is a separate obituary, a small article. It takes reading thru the whole paper a few days before and a week after the death date. Yes, I have found incorrect death dates on death certs.
The other major tip is to research every relative you can find. This has done it for me a number of times. Look at censuses for neighbors who might be relatives. Build up Ancestry pages on every member of the family. I found the origin of one gg grandmother thru a listing in a tiny surrogate's court legal ad.
I'm sure you know the Brooklyn Eagle, Brooklyn's main newspaper, is free online thru 1902. Many later issues are available free thru fultonhistory.com.
But I too still have several Brookly gg grandparents who are total mysteries re their parents' names or origins. Always glad to hear other suggestions.