I have a friend with a similar situation, although he is not interested in dual citizenship. He lives in South Africa and both of his parents were born in Germany (Berlin) approximately during the first world war. His grandparents were all born in the Austro-Hungarian Empire. His parents left Germany as adults in 1939 and moved to South Africa.
He asked at the German consulate for information about German citizenship. He was told that, at the time, Germany did not recognise birth as automatic citizenship, as some countries do today. This means his parents needed to apply for German citizenship before they left Germany. His family is Jewish, which, of course, makes this even more complicated since it is highly unlikely the Nazi government would have given German citizenship to "foreign" Jews.
The point is that without German naturalisation papers, birth in Germany during that time is simply not enough. In the end, my friend gave up.