Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object represented by a point of some similar moduli space (sometimes even of itself?). Why on earth should this happen? How such a strange possibility could possibly occur to anybody?

And yet I know this is one of the most deep and important achievements in current mathematics. Can it be given some sort of heuristic/intuitive justification an outsider could understand?

Proper reaction to such questions probably is "why don't you read this or that paper"? However I would only accept such answer if the corresponding paper contains a good noob-friendly introduction since I certainly am a noob in this case.