Thank you everyone for all the responses on this thread.
I did put a photo request on findagrave, but a fellow researcher who lives close to the cemetery in NJ, USA reached out to me off list in response to this post (or possibly in response to my post on the Jewish Genealogy Portal on Facebook), and after a few visits to this unfortunately neglected cemetery and with help of another neighbor of hers who is involved in the cemetery, she was able to locate and photograph the graves of Harry and Rose, and three of their children buried there. Although on gravestone of Harry and Rose we have the Hebrew dates of birth and passing, we do not have their Hebrew/Yiddish names with their patronymics. So I am unable to know Harry's father's name from this. On gravestone of daughter Dorothy, we have both Yiddish patronymic הערשל (Hershel) for Harry and Hebrew matronymic רבקה (Rivka) for Rose. These are the names that also appear (in English letters) on some of the immigration documents, etc.
So, I uploaded the photos to finadagrave, and If anyone is interested, the link for Harry is:-
(From there, there are links to the gravestone memorial pages of the three children whom I mentioned.)
I wish to add, that on what very much seems to be my third great-grandfather's gravestone in Bialystok (which resonates with my oral family history and also resonates that this is after whom my grandfather was named):-
the transcription mentions (in Polish) that he was originally from Brisk (Brest), which resonates with the known family history of the ZIENTS family that is the focus of this thread.
So any further help in trying to find out who Harry ZIENTS' father was, would be welcome.
Ma'aleh Adumim, Israel