Ok, let me start with the obvious. No, you cannot go swimming in Grand Prismatic Spring or any other geyser in Yellowstone National Park. While it is indeed illegal, the truth of the matter is that most of the natural geothermal features of Yellowstone are just a few degrees shy of boiling water, subjecting the would-be bather not just to federal law, but the law of physics, and near certain death :)
That said, I distinctly remember that in Yellowstone, there are some places in the rivers that you can actually swim and experience a hot springs like effect. Where the runoff from the geothermal features meets the river (like Firehole), the mixture can be quite pleasant.
I am bringing my kids to Yellowstone in August, and I want to take them there. Unfortunately, within my books, I can't seem to remember where one can do this.
To the best of my recollection, on the way up from Norris to Mammoth Hot Springs, there is a place where you can pull off on the side of the road and get in the river, where the runoff from a hot spring naturally warms the river. Does anyone know the name that I could google to do more research?