There is an old notion of parodia, meaning "reworking an existing piece of music for a different purpose, usually with unrelated lyrics". Together with the practice, that notion came out of vogue about 1800. Therefore, the English word "parody" should only be used in the modern sense.
However, the goal of modern parodies is not always to satirize the original author or genre. Many parodies preserve the melody and some key phrases of an existing song, and put them in a different context. I would use the term "parody" only if the contrast to the (well-known) original lyrics is an essential part of the message. Otherwise it should be called "recycling".