~If you are really interested in "settling" the dispute, and being able to justify your position, why are you asking here? Punch up "Arab", "Persian" and "Iran" into a search engine and do a little (actually very little) reading. Once you figure out how you are going to define "Arab" that is, by genealogical, cultural or linguistic categorization, the task will be much easier. The etymology is important.
Of course, if the definition you select will include Sumerians, Akkadians, Babylonians, and Assyrians as 'Arabs', you may complicate the issue. I suspect one could classify the Jews as Arab on the basis of some definitions.
Then too, you have to decide what constitutes a Persian. Do Aryans (Proto-Iranians) count? How about Indo-Iranians? Just for fun, throw the Oxus civilization discovered at the Bactria-Margiana Archaeological Complex into the mix and decide if they were forbears of the Persians and, if so, whether or not or not they can or should be called Persians.
Modern Iran was called Persia until 1935. Does that make a contemporary Iranian a close relative of Darius, Cyrus or Xerxes? If so, I guess that makes Iranians and Persians the same thing. By that logic, I guess the Russians and Danes and English and French and Italians and Irish and Dutch and Spanish and Portuguese etc. should still be called Teutons.
Islamic or Muslim? Give me a break. During the classical period, Persians were Zoroastrian. Before that they had other beliefs. As with all civilizations since the dawn of time until the present, mankind creates the gods he needs and then builds the necessary religion around those gods to suit the required purpose - generally to concentrate the wealth and power and to keep the masses in line with superstition and myth. It worked then, it works today. "Religion does three things quite effectively: Divides people, Controls people, Deludes people." Mary Alice McKinney "Faith is the great cop-out, the great excuse to evade the need to think and evaluate evidence. Faith is belief in spite of, even perhaps because of, the lack of evidence." [Richard Dawkins]