I have always enjoyed the idea of creating "parameter spaces" or "moduli spaces," but it is only recently that I have seen very concrete applications of studying the moduli space. Because of how pervasive this theory is, I was hoping that
Notable examples I have come across:
enumerative geometry in the sense of trying to solve a geometric problem by replacing it with intersections of submanifolds (varieties) of the moduli space. This is pretty similar to an idea I had here although I don't know if this approach is fruitful at all.
Complex dynamics. In particular how you can figure out some properties of a quadratic polynomial by looking at the mandelbrot set.
Vector Bundles Defining the Euler class via the interpretation of $\mathbb RP^1$ as the moduli space of lines in the plane.
Are there other applications of moduli space that solved a concrete problem? I included (3) mostly to exhaust my knowledge of the subject, but I view that as quasi-concrete in the sense that understanding the topology of the moduli space can be used in a serious way to classify line bundles.