I must admit Parr is not a name I am interested in because I never considered it to be Greek.
Unfortunately, American Greeks, although most of them are very proud to be of Greek origin, some completely "destroy" their names as if they try to hide something or do not want to be found. The last 10 days, I downloaded about 500 obituaries from USA and I must say, some of these changes do not make me very happy and I do feel sorry for people like yourself, trying to find who they are but have nothing to work on. I also have lost relatives in USA that I can not find, but I do not give up.
Now lets check the facts:
Names: Virginia is not a Greek Orthodox name, the Greek equivalent is "Parthenos". Robert is not a Greek name and has NO Greek equivalent.
George & Kathlyn are used by Catholics and Orthodox. Lucille is not a Greek name, the Greek equivalent (Translation) is "Loukia" and is never used. SO the first names indicate Catholic Anglo-Saxon back ground.
PARRHIMENOLOUS: It consists of two parts: the prefix PARRHI and the first name MENOLOUS.
In Greece we have the following names that start with the letter "M". Manolis, Manoussos, Menios, Manthos, Markos, Menelaos, Minas, Minos and Markellos.
From all the above, "Manolis" is the closest to Menolous. So, the second name is either Manolas or Manolou if it refers to a women.
The prefix "PARRHI" does not exist: Greeks use the following prefixes: Papas, Pappas, Hatzis, Karas, Mavros, Kontos, Maestros (Mastro), etc,The Greek language has not names or words that use two "R", and in translations, the letter "H" rarely follows another Greek letter with two exemptions, the letter "R" as in "Rhodes", "T" as in "The" and "C" as in "Ch". So, It does not matter how you looks at it, nothing fits.
A name that comes close, from unknown origin, but has nothing to do with "PARR" is "PARREN.
Kalliope Parren, 1861-1940, born in Rethymnon, Crete, was the first Women activist in Greece, pushing the importance of Women in society as the published and editor of her women magazine.
Now have a look at this: English letters: Pappas. Greek letters: Παρρας
The only difference is the first Capital letter "P". the Latin small "p" and the small Greek letter "p" are identical.
This mixup is obvious when you check the names "Paraskevopoulos", one person shortened it to "Parras" the others mistook the Greek "ρ" for English "p" and shortened their names to "Pappas".
There is an answer to everything.
The ionian Islands, for numerous years and around 1840-1880 (Check history) were under British control. There were many British families of Officers and their relatives in Corfu, Zante, Lefkas, etc.You will find your family among those officers or supporters of the army. It is possible that one (Robert) married a Greek/Venetian lady and had a son named George who had a son named after his father Robert.
The person you are looking for is Robert Parr, born around 1850 in Corfu (or Ionian Islands),
The question you must ask is: Who from the British Parr family, was a soldier and served in "Greece" around 1850?