A parasol is a fashionable 19th-century object used to block a woman's face from the sun. But let's face it—it's mostly a fashion accessory. Let's just say if you were an "it" girl a couple centuries ago, you'd be hoping for a parasol named after you—not a handbag.
Daisy Miller never leaves home without a parasol, even though it's often folded up. So we associate Daisy with her parasol, and it comes to symbolize many things. Most of all, it's about flimsy or superficial protection between her and the dangers of the outside world.
Daisy's out on her own with strange men in Rome, and all she has is this thin, white umbrella that isn't even waterproof. It's like driving with a paper seat belt or riding a bike in a clay helmet—if either of those things for some reason became fashionable.
When we first see her, she's wearing no hat, but has a large parasol. The 19th-century hat is a way of emphasizing one's propriety, respect, and station. Your taste in hats lets everyone else know what you're about, and hats are used to defer to others "I take off my hat to you!" or to display honor (think of a man taking off his hat to meet a lady.)
Daisy is bare-headed—gasp! You can imagine what that means. But her parasol is very large. This seems to imply that she lacks respect for others and that her own protection of herself (her face) has been replaced by something that is much more for show than for practicality. Her parasol is fancy and embroidered, and often she carries it with her closed. At one key moment, she even allows Mr. Giovanelli to handle it for her:
The western sun in the opposite sky sent out a brilliant shaft through a couple of cloud bars, whereupon Daisy's companion took her parasol out of her hands and opened it. She came a little nearer, and he held the parasol over her; then, still holding it, he let it rest upon her shoulder, so that both of their heads were hidden from Winterbourne (2.136).
The parasol should be shielding her from the heat of the sun, but instead it's directing her right into the heat of Giovanelli. The very thing that is supposed to protect Daisy from the elements is now used as a tool to further her own exposure.
Because of the parasol, Winterbourne doesn't have a clear view of Daisy and Giovanelli. He has no idea what they're doing behind there. And what he's imagining is probably way worse than what's actually going on. The use of the parasol as an ad-hoc privacy screen is a moment that neatly summarizes Daisy's lack of awareness of (or indifference to) the way her behavior reflects on her.