It's not unusual to bring along an accessory to enhance a dramatic skating program. A hat (top or beret), a scarf (or two), a cane (or a baton), an umbrella (or a parasol), a twirly ribbon, an Asian silk folding hand fan, a fake microphone (or a megaphone), I've seen all of these. Sometimes it seems that the skater brought the prop along just to support the mood already established with the theme of the music, but I've seen a couple of devices that when used actually do add a bit of glamour or some other interesting flavor to a skater's elements.
I can't say though that skating with an umbrella or parasol adds that much to the panache of your performance. Unless you can manage to spin it in the opposite direction over your head as you execute a fast spin (and good luck with that) a parasol just doesn't have much graceful glamor to it. Plus skating with an open umbrella seems like a particularly difficult task, as the air pressure provides such uneven resistance depending on your direction and the umbrella's tilt.
Skating with a cane or baton is easy enough, and a good little trick is to attach a leather band to the handle to make it easier to twirl. If you're going to carry a cane about the rink then really use it though! Tap it here and there, twirl it some, use it as a pivot in a move. Please though don't throw it into the air to catch it. I can't even begin to tell you how many times I've seen the timing on a program royally screwed up by a dropped baton. And even if you can toss and catch it 90 percent of the time during freestyle, your chances are only about 30 percent during the competition. And if you set it down for some other elements just be sure that you don't lose it in the process -- back gliding over a dropped cane can result in a fairly serious injury!
Many times I've seen somebody skate with one or two small collapsible accordion silk hand fans. You know, the kinds with drawings of Asian bamboo or flowers or dragons. Fans add quite a bit of expressiveness but it takes a bit of practice to control expanding and collapsing them effectively, especially when you're on the move. They are also easy enough to tuck into an elastic band to secure them in place when you want them hidden (stage trick: Velcro).
Skating with hand-held scarves or gauzy trailing things can create some nice visual effects (you know what I mean, those ribbons of lightweight floaty fabric that you hold in your hands or tie to your wrists). I'm not talking about the ribbon that gymnasts are fond of using for their routine, a long narrow tape on a stick. Although you can use one of those too if you'd like. The real issue with skating with a gymnasts ribbon is that to be effective you really need to keep it moving the entire time, as (aside from dropping it on the ice) there's no convenient way to get the thing out of the way. So that means if you use one, your entire routine is pretty much about the ribbon.
Most popular though seems to be ribbons or swaths of gauzy fabric, but these though aren't without their own problems. Solid props are much better behaved as they go where you place them; anything with fabric though tends to have a mind of its own. This is especially a problem on any kind of spin. I was at one event where a skater managed to unintentionally wrap a rather wide chiffon around her head during a spin, and then had to grapple it away before moving onward. If you're going to skate with something gossamer you need to plan ahead for every spin and jump and decide how you're going to secure the thing, either with some attention to wrapping it around a body part, temporarily scrunching it, releasing it for later, or holding it far enough away that it doesn't become an issue.
Finally, one of the nicer effects I've seen is to attach a somewhat wider swath of chiffon to the tines of a fan. That way you get a more controlled and sustained width of fabric fluttering about. Again though this makes stowing it away a bit of an issue.
Whatever prop you use, plan the trajectory of its use carefully across the length of your program, and be sure to use it in a way that really adds value to what you're trying to present.