I certainly didn't see the episode but apparently this is real. Big Bird asked Michelle Obama "Maybe you’re here to finally show us your husband’s United States birth certificate", Michelle's says no, and then Big Bird say's "That's because you can't!" I need to investigate this further but watch the video. It starts slow but it is worth it.
Apparently this is a parody made to be funny. So laughing at the fact that one can usurp the office of the President is now OK?