As I remember 'parody' doesn't have to be humorous, although nowadays it often is. When a parody is taking a rise out of the original song that is 'burlesque'. To be a parody all it needs to have is the structural description you've already given, i.e., that the new song/poem is using the original song in some way, some of the text, the basic structure, and/or its tune, and it should be at least obvious at the time it is written what the original song was.