Welcome to the RTA Forum!

Let's start by saying that trying to drive from Portland to Yellowstone/Gardiner "straight through" is an incredibly bad idea. At nearly 850 miles, that's going to take you a minimum of 15 hours once you factor in just the basic stops like fuel, food, etc. As a one day trip with multiple drivers, you might be able to make it safely, but it is going to wipe you and your family out, and will make it difficult for anyone to enjoy the trip for a couple days afterwards. It would be much better to plan a more reasonable first day, aiming for someplace like Missoula, where you would have a good 10 hours on the road, but could also stop or two and enjoy the drive. At the absolute farthest, I'd look to stop at Butte.

As far as water parks and amusement parks, Yellowstone and nature is the attraction of the area, and I'm not aware of any sort of man-made parks like you are talking about. That said, there are lots of ways to enjoy that nature beyond siteseeing. There are several whitewater outfits in the area. We loved our rafting trip on the Snake based out of Jackson, but there are also outfitters for the Yellowstone River in Gardiner. You could also look at horseback trips, and there are some places both in Yellowstone and the Tetons where you can do some swimming.

Make sure to check out the RTA Map Center and other parts of this site for specific ideas about places and ideas for things to do both around Yellowstone, and for your trip both there and back.