Quoting ma ♥:" We go to theme parks a little more than occasionally, we go just about every other weekend. If we're ... [snip!] ... don't bring a stroller with you? & All of the umbrella strollers I've seen have a lower weight limit than what he weighs."
My 2 year old has a Chicco Liteway which is an umbrella stroller, has a basket, and lays down. I keep that in the trunk. It was very cheap, $140 I believe.
Instead of getting another stroller for your preschool aged child I would work on building up their stamina to handle all the theme park trips you take. There's really no reason a 4 year old shouldnt be able to keep up. Especially with so much down time on rides, shows, eating, etc..