In the beginning, we had a plot of yields of individual bonds against time to maturity, the crudest form of "yield curve."
Years later, people began hand-drawing a smoothed line through these yields as closely as possible. Because bonds have different coupon rates, making their yields hard to compare, people tend to draw the curve through bonds trading close to par (100), making these the earliest form of "par yield curves."
Later on, people discovered that they can use much better models to construct these curves more "scientifically" (e.g., in ways that use discounted cash flow principles and account for the differing coupon rates of bonds). Many models are proposed, including the Nelson-Siegel model, cubic splines, exponential splines, etc. These models all attempt to accomplish the same thing – create a curve that best fits the prices or yields of observed bond yields/prices.
Now back to your questions:
Nelson-Siegel, like any other curve fitting procedures, can be used to produce smoothed yield curves. The outputs from the model can be the zero coupon curve (zero coupon rates against time), par curve (yields and coupon rates of par bonds against time), or forward curve (forward short-term interest rates). These curves are just mathematical transformations of each other. From a single model, you automatically get all of them.
The inputs to these models (NS included) are almost always coupon bonds, not zero coupon bonds (so yes, bonds of different coupon rates are used to calibrate the model). But as said, you get both zero coupon curve and coupon yield curve out of the model.
Quants/researchers like to work with zero coupon rates, because of their mathematical simplicity. But par curves are frequently the preferred presentation format, since they are more directly comparable with observed yields.