When is a solution to a multivariate polynomial not parameterizable?

Question

This is a follow up to a comment made on one of my previous questions that I feel makes more sense as a separate question. The comment was to the effect that the real solutions to a system of polynomials may not be parameterizable unless the system is "nice". When would a solution not be parameterizable? Could I not just choose arbitrarily some of the variables to be parameters, then solve the remaining system? Or is that too naive?