Well, as far as my history knowledge goes, the term "Palestine" for the region was introduced by the Romans. I thought, that was more well known? - Here, quoting from Wikipedia (https://en.wikipedia.org/wiki/Palestine_(region)):
Roman authorities renamed the province of Judaea to "Syria Palaestina" in c. 135 CE to punish Jews for the Bar Kokhba Revolt and permanently sever ties between Jews and the province.
Ah, right. The article also states that the term has been used by the greek Herodotus in the 5th century BCE. So it was probably known and used some time before the Roman conquest of the area, I suppose?
So I guess, the statement from your parents seems mostly right. Maybe just framed bad. Unclear why this is a Zionist take, if it's just the documented history.