I can't fully explain the passion for parasols. Shape is part of it, that regular gentle curve that leads the viewer's eye to just where the artist would like it to go. The potential for unexpected color is another. A pop of orange can be excellent when at sea or in a painting of a grey, rainy day.
Umbrella advantages can be subtle as well. Faces shaded by umbrellas are often lit by soft reflected light with shadows that take on the color of the umbrella itself. The most famous painted lady with a parasol, Monet's wife, is such a mix of shadow and reflected light that she almost dissolves into the sky and clouds behind her.
My lady is warding off the last drizzle of a Manhattan shower; there's really more sun than rain in her forecast.
NYC Shower 8x8 oil on panel.