Palestinians are not Semites and Jews aren't Semites either. That is 1800s race theory out of Germany, the same place and time that coined the term anti-semite as a source of pride. And the only "semites" the Germans knew living among them were Jews. Still, like many archaic terms, the term antisemitism has stuck with us but means simply dislike, aversion and hatred of Jews.

Jews, Palestinians and Arabs in general are part of the rather useless categorization of caucausoid along with Europeans. That is as far down we drill on race in anthropology these days. In biology there is only one race.

In linguistics there is the Semitic language family which includes Hebrew, Arabic and many north African languages.

The fact the ethnic group now known as Palestinians and the ethnoreligious group known as Jews both, for the most part, have genetic ties to the Levant(Palestine and surrounds) and are closer to each other than they are other groups (with groups like Kurds and Armenians plus other Arabs being the next closest) is an important point. But we should use correct language and not call either Semites, who don't exist.

Even the words racism and racist aren't really correct and the meaning has changed over time to reflect animosity towards ethnic/ethnoreligious groups rather than "race" ( because our understanding of race has changed making old classifications reduntant).

But the issue of saying they are all Semites is that many Palestinians and other Arabs take up this argument to suggest they are either "the real semites" ( ergo Jews are "fake semites") and that their comments can't be antisemitic because they are Semitic themselves, when they display hatred of Jews.