義母：　　のりこさん　車運転できましたよね
のりこ：　エエ　あまりじょうずではございませんけど
義母：　　…乗せていただけないかしら
のりこ：　そこで待っていてください　今　車を持ってまいりますから
のりこ：　車を乗せていただきたいと聞こえたものですから
義母：　　まァ　そうだったの
義母：　　私は車に乗せていただきたかったのよ　をとにの違いでこんなに違うんですもの　日本語ってホントにむずかしいわね

The best part is, of course, that neither particle's used in the original request.

I get the gist, but I await further commentarious enlightenment.

OK, as far as I can tell, it gets its "humor" from leaving out information in the original request, leaving normal Japanese readers to assume something, and then something different happens because the woman is insane. Har har har.

Very lazy of the writer, as far as humor goes. But quite common in English-language humor as well. Or am I missing something?

*joke spoilers*
車に乗せていただく＝Please let me ride in the car
車を乗せていただく＝Please put the car on top of me.

Also, can you imagine what Tom and Jerry would be like if they were this polite to each other?

トム：あら、ジェリーさんはとてもおいしそうになってますねー。
ジェリー：いや、お違いになりますよ、トムさん。そんなにおいしくございません。わたくしはとても細くございますので、私の体を召し上がってくださればお口にあわないと思いにいたします。
トム：しかし、それにしても、ジェリーさんを味したいとおもいますよ。
ジェリー：ほんなら、わたくしが逃げにいたします。
とむ：さようなら。体お大事にしてください。

Damn, Denske, you harsh!

OK: The book is a series of vignettes, about that long, in which Noriko tries to do her mother in and her mother narrowly escapes death (or fails to die even after suffering what should be mortal injuries). It really is like Tom and Jerry (or maybe Itchy and Scratchy), except with the added background of an ageing Japan where the older generation expects to be taken care of by their kids when they get too old to live by themselves (like they did for their parents and parents-in-law), and the younger generation is feeling like that's a Japanese tradition they could do without.

The whole を/の thing isn't really meant to be funny in and of itself -- the humor is basically two running gags that occur in every vignette in the book: (1) the incongruous cartoon violence (come on, that panel at the top of page 4 is funny), and (2) the part at the end where Noriko makes an obviously false excuse for her behaviour and her mother-in-law pretends to believe it.

(I see now that this is impossible to get unless you've read a few others for context. Sorry about that. Maybe I should have just put in the panel at the top of page 4.)

Also, anon-0 makes an excellent point. Maybe I should have called this "a cautionary tale about omitted OPs".

Sounds like Noriko is trying to bring back the even older Japanese tradition of obasuteyama.