By "umbrella" you seem to be thinking only of a
WalMart cheapy for under $20. You need to work on widening your definition.
For travel of any kind, an
umbrella-fold stroller is always going to be your best option. They are lighter and easier to carry and stow, and have fewer dangling bits to get snapped off in transit.
The features that you really need for theme parks are: 1) tall handles that won't kill your back when you are pushing it for hours on end, 2) recline, 3) a good-sized hood for shade, and 4) good suspension. I cannot emphasize enough how important good suspension is. An indicator of a stroller that has it will be the presence of double wheels on the back, though there are levels of good. (Macs have the best suspension on the market, which is reflected in their pricing.) Good suspension enables a stroller to still roll freely even when carrying a heavy load; much less work on an all-day outing.
Umbrella-fold strollers can easily be carried with one hand and are much easier to take on trams and the MK train than square-fold models. All but the super-cheapies normally will have a small basket and perhaps a hood pocket.
You can always add a cup holder and storage; do NOT let those be your criteria for purchase. (I use this cupholder on all of mine:
http://www.amazon.com/Sunshine-Kids...ie=UTF8&s=baby-products&qid=1235156886&sr=1-3)
Using a large basket while in the parks is not a great idea anyway; you will be tempted to bring too much, and it will a major PITA to empty it all out to fold the stroller for transport on buses and trams. Put most of your gear in a backpack and drape it over the stroller handles so that you can just grab and go when you park outside of rides.
Some choices: Maclaren, Peg Pliko, Inglesina, Chicco, Baby Planet, Rock Star Baby, Mia Moda, and Joovy. There are also a couple of models from Graco and Evenflo now, but they tend to be made of steel, so they are a bit on the heavy side for minimalist designs. Combis are also pretty good as long as neither you nor baby is overly tall (though Combis do not actually umbrella-fold, they are light and compact.) Kolcraft reclining models are the baseline in this market segment; these are the ones sold in WDW stores.