Was there Palestinian involvement during World War 2?

Yes

It is unclear what is meant by "Palestinian Involvement". The Jewish Militias of Palestine, like the Haganah and Palmach (who at that time were called Palestinian Jews) actively contributed to the British War Effort against the Nazis. The Palestinian Arabs did not assist the British War effort, although they were involved in supporting the Grand Mufti of Jerusalem who was a close friend of the Nazis and Fascists in Europe.