No, "Rice Rice Baby" is not a Weird Al Yankovic song. The song was produced by Mark Jonathan Davis. On his official web site, it says:
In 1990, [Davis] moved to Los Angeles to manage the Premiere Radio Comedy Networks, where he produced parody songs including "The Star Wars Cantina," "Jeannie's Diner," and "Rice Rice Baby."
Another web site maintained ...
Yes, it is a cover/parody/translation of the song "Mirror" by Lil Wayne ft. Bruno Mars
He pretty much just translated all the words form the original song into Yiddish (unlike what he did with "Bas Kol")
YouTube video of Mirrors
It may be a bit uncomfortable to watch that video, here is a link for an audio-only video with lyrics.
You ask two questions here.
To the question : "are there any musical parodists who...", there is this compilation video from Helen Arney and Jay Foreman : where they "explore common meter (or "common metre", "ballad meter") by mashing up a whole lot of songs."
For wider context, the technique is referenced under TV tropes as common Meter and more ...