I have a way of generating random parameterized maps from $S^1 \to \mathbb{R}^3$. This method can create very simple knots, such as ellipses, but can also create knots with more crossings than I can count. The images can be knotted as I have seen the method generate a figure eight knot. I know that with a parameterization instead of a diagram one can compute things such as various knot energies. My question is the following: Is having the ability to generate random parameterized knots of use to anyone?

Not very likely. Any physical questions about random knotting will come with all sorts of requirements about the random process that generates the knots, and so it's unlikely that whatever method you have in mind would be helpful in any given situation. Also, it's very easy to produce random knots, and without specifying some nice property your method has it's hard to say much. The simplest method I can think of is to pick n random directions in R^3, and concatenate a series of n unit intervals in those directions, then join the first and last point. Why should your method being any more or less interesting than this one? You might be interested in reading Dorothy Buck's papers on the arXiv, which include some simple models for transformations of knots that are relevant for knotted DNA, to get an idea of the very particular requirements of specific applications of "random knotting". 

