The narrator, expressing the thoughts of Mrs. Spragg in Chapter 1, frets that Undine isn't making anything of herself in New York and has received no "social benefit" from the city:
[Undine] seemed as yet––poor child!––too small for New York: actually imperceptible to its heedless multitudes, and her mother trembled for the day when her invisibility should be born in on her. [...] [Mrs. Spragg] had noticed lately that Undine was beginning to be nervous, and there was nothing that Undine's parents dreaded so much as her being nervous.
This worry is in response to the fact that the Spraggs moved out of the Midwest explicitly to improve Undine's social prospects: "They had left Apex because Undine was too big for the place." But now Mrs. Spragg is worried that Undine is invisible and small and nervous—to a hyperbolic degree. Note the certainty in Mrs. Spragg's concern: Undine is "actually imperceptible." There is also great hyperbole in her concern for Undine's "nervousness." It is quite the overreaction to "dread nothing so much" as nervousness, a relatively normal feeling to have. "Nervousness" often had a stronger connotation at the time, referring to a more serious mental health problem, but Mrs. Spragg's worry is still excessive. Mrs. Spragg's worry about Undine is a hyperbole that helps depict the mother's character. Her overreaction shows how fully Mrs. Spragg lives vicariously through her daughter.
This overreaction is also an ironic foreshadowing of the rest of the novel. Far from "invisibility," Undine, at every turn in the novel, works to make herself more visible, always wanting to be in the presence of adoring onlookers. Mrs. Spragg, ironically, worries about the precise opposite of what she ought to be worrying about. This irony serves to further characterize Mrs. Spragg as vapid and as an inaccurate judge of her daughter's character.
Mr. Spragg and Undine discuss the party at the Fairfords in Chapter 2, cracking jokes back and forth. Undine, just before the passage below, has explained that Mrs. Fairford is related to the Dagonets, a wealthy family. Mr. Spragg snaps back:
"What do they want to know you for, I wonder?" he jeered.
"Can't imagine—unless they think I'll introduce you!" she jeered back in the same key, her arms around his stooping shoulders, her shining hair against his cheek.
"Well—and are you going to? Have you accepted?" he took up her joke as she held him pinioned; while Mrs. Spragg, behind them, stirred in her seat with a little moan.
This sort of back-and-forth is typical of Undine and Mr. Spragg's relationship, as they make ample use of verbal irony. There is truth at the core of their jokes, though, and Undine speaks in "a tone between banter and vexation." Their jeering is, it seems, good-natured.
But the ironic moment (when Undine quips that the Dagonets must just be using her to get closer to Mr. Spragg) does point to an important feature of the Spragg parents throughout the novel. Mr. and Mrs. Spragg are as interested in Undine's social prospects as she is. This is due to familial affection, of course, but also due to personal interest. The Spragg parents want to ensure their connections in Fifth Avenue society, both socially and financially, so they live vicariously through Undine. While Undine and her father joke back and forth, they subtly evoke this tension in the family that carries throughout the novel.