From the my limited experience it seems that often a parameter space (miraculously) has some of the properties of the elements it parameterizes. For example:

- The parameter space of all plane conics is an algebraic variety.
- The moduli space of genus $g$ curves $\mathcal{M}_{g}$ is itself an algebraic variety.
- The Mandelbrot set is connected and can be thought of as the parameter space: $$\{c\in \mathbb{C}\; : \; J_{f_c}\; \text{is connected}\}$$ where $J_{f_c}$ is the Julia set of the polynomial $f_c=x^2+c$.
- The Teichmuller space parameterizes complex structures on a surface $X$ and itself is a complex manifold.

I am wondering if there is a reason for this occurring. State a bit more formally my question is:

Question:Why is it the case that if parameter space $M$ parameterizes spaces of with property $P$ then $M$ (often) also has property $P$?

Note I say often as there are examples where this does not happen. For example, there exists quadratic polynomials whose Julia set is not locally connect and there exists quadratic polynomials whose Julia set is locally connected, but of course the Mandelbrot set is either locally connected or not.