The problem with the night parades is that there isn't a lot of light and it is moving. For this you need to combine a high ISO (typically at least 1600) with a wide aperture (typically f/2 or wider) to be able to get a fast enough shutter speed to capture the light AND stop the action.
This is why you'll see a lot of suggestions for a lens like the 50mm f/1.8. f/1.8 is a very wide aperture (keeping in mind that f/stops are fractions so the smaller the number the bigger the aperture opening) and is a great tool for this type of photography. Its also a very inexpensive ($120) lens. The issue you will run into with this lens is that it does not have a focus motor in the lens and thus will not auto focus on the D60.
Now, another option is the recently released Nikkor 35mm f/1.8 AF-S. It is wider, but overall might be a better fit for Disney low light parade/indoor ride photography. It is a little more expensive than the 50mm f/1.8, but still relatively inexpensive for a lens at $199. It does have the focus motor in the lens (the AF-S lettering tells you it has the focus motor in the lens) and will work perfectly fine with the D60.
Another advantage of prime lenses is that they are typically sharper than zooms. You'll get greater depth of field using that large of an aperture. Keep in mind that you don't have to just use these types of lenses only at f/1.8. They are very sharp between f/2.8 and f/11 and are a great general purpose lens. I have the Sigma 30mm f/1.4 and will leave it on the camera for long stretches just for general candids around the house. Very sharp and very nice image quality.
Camera's and lenses are tools. Not all tools will do all the jobs you want.