"Av harachamim" is a prayer asking God to recall those who died al kidush hashem. It was written in response to the First Crusade (and is thus said only in the Ashk'nazi rite). Because of its somber topic (or origin), it is omitted on various joyous occasions, though which occasions qualify varies widely. It is also, according to some views (cited in Mishna B'rura 685) and in many synagogues, omitted when the "four parashiyos" are read. Why omit it when the "four parashiyos" are read? That's not a particularly joyous occasion.
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First of all, it should be noted that there is a custom to say "Av Harachamim" on the four Parshios-Shabbosos as well. The custom not to say it is in the Darkei Moshe in the name of the Maharail (Siman 685), and seems to be the custom of the Magen Avraham and R. Yaakov Lorberbaum of Lissa (in his Siddur, Derech Hachayim). However, the Eliyah Rabba (685:18) and Shaarei Efraim there mention the custom to say Av Harachamim even during the 4 parshios.
However, the Aruch Hashulchan (685:9) seems to approve of the custom to omit the prayer, because these Shabbosos have a festive air about them, just like the Shabbos Mevarchim.1 It sounds like anytime when something special is done regarding the leining, there's a feeling of happiness (because there are more kibbudim to be given out if there's another Sefer Torah, maybe? I'm not sure).
R' Yaakov Emden (Mor Uktziah 685:10) writes that once Adar has entered, "מרבים בשמחה", we increase our joy, and don't want to dampen that joy with the somber prayer of Av Harachamim.2 (Even though there are week(s) in Adar without an extra parsha, I assume that this isn't an issue because the festive nature of the month isn't express in any halacha during that week/weeks)
My understanding has always been that these parshiyos are related to the holidays around which they are centered (Purim and Pesach), and so the season, at least on a Shabbos when that feeling is being expressed by the "four parshiyos", should be a joyous one.
1 "ולעניות דעתי נראה שלא לומר כיון שאין מזכירין נשמות והם שבתות של שמחה ולא גריעי משבת שמברכין בו את החודש"
2 "במגן אברהם הביא בשם מהרי״ל שבכל ד׳ פרשיות אין מזכירין נשמות… לא נודע לי טעם בפרשת שקלים אבל בפרשת זכור שכבר נכנס אדר שמרבין בשמחה נ״ל שנכון מנהג מהרי״ל וכ״ש בשתי פרשיות שלאחר פורים שהם בין גאולה לגאולה דהוי מנהגא"
Those that omit Av Harachamim do so because the Four Parshios are to some extent celebratory, in that they usher in and remind us about the oncoming festival of Pesach. The poskim failed to make this explicit because they mistakenly presume that the reader would realize this on his or her own. Of course, my assertion is, by its very nature, difficult to prove.
Mishna Berura 685:18 actually brings a dispute as to whether we say Av HaRachamim during the weeks of the 4 Parshios. Minhag Chabad is to say on Parshas Zachor and on Parshas Para מפורש בלוח 'היום-יום' לש"פ זכור ופרה שאומרים 'אב הרחמים', however do not say it on Parshas Shekalim or HaChodesh since we are either Bentching Rosh Chodesh or it is Rosh Chodesh.
I have yet to see a reason mentioned anywhere for those that do not say it - other than "we do not say it". My own thought is perhaps there is increased Simcha in taking out a second Sefer Torah.