Third Person (Objective)
Parker doesn't really go into her characters' heads—the story is almost entirely comprised of dialogue, except for the beginning paragraphs and the occasional stray sentence. Parker does make a slight judgment at one point, though, writing, "eight minutes for the settling of two suitcases and a hat-box is a long time"—meant to indicate the husband is avoiding talking to his new wife and hashing out the details of what's going to happen in their marriage (bed).
Aside from that one little tidbit though, the presentation is entirely objective, straightforward, and judge-for-yourself. Part of the humor of the story comes from figuring out what, exactly, the characters are tiptoeing around, and the objective third person perspective helps in creating this puzzle.
It's a little like the Hemingway story, "Hills Like White Elephants"—only, in "Hills Like White Elephants," the main characters are (spoiler alert) discussing whether to have an abortion or not, and here the characters are (implicitly) worrying about consummating their marriage.