There are a few approaches to this question:
The oldest continuously operating theme park is Cedar Point. But that's a little dubious, as it was really just picnic grounds and a recreational center in the 19th century. It has had many coasters come and go in the early 20th century, but when Blue Streak was built, it was the only coaster at the time. It wasn't really until the last 40 years that the park really became anything of importance. Really, Magnum is what put it on the map. I believe it may have previously been Geagua Lake before it was razed.
Technically, you should give it to Disneyland. Amusement parks, fairs, expos, beachside parks, etc. were all the rage, but Walt Disney really coined the idea of heavily themed rides with stories and sections of the park with specific themes.
Kennywood has an interesting claim. The way that they put it is a little misleading, but it probably preserves the old timey feel more than any park in the country, and has the oldest operating rides. But that's not because its been around forever, just because that's the theme that they're going for.
Honestly, in business, I'm not a big fan of calling attention to your age... unless you're a household name and industry standard. My philosophy is that if your business is incredibly old and it is still regional, its been a bit of a failure, and you shouldn't call attention to that. Unless you've gone national, you should try to re-brand every so often so appear fresh and to show that you're on the rise.
And Kennywood has a unique problem as well. By advertise their tradition and age, they're discouraging visits. They're telling guests in other states, "Don't bother visiting now, we'll still be here decades from now, so come whenever you want, we'll be waiting." Then the guests never come. You want to create some urgency, where if you don't come now, you might miss out on what we have now.
but that was my whole childhood)
