No, it's not like a comical version of the Roman Catholic Mass...
The relationship between religion and music has always been a mildly dysfunctional one in western civilization. While, on the one hand, you can thank the Christian church in Europe for repeated whitewashings of culture, beginning most fabulously with the elimination of music of pagan Rome, the Church must also be credited, for we would not know the great heights of beauty now known as pop music were it not for the tradition of chant, which itself has its roots through the very earliest days of the Christian faith as an established religion.
One of the distinctive elements of Western music is its use of polyphony, and it is in religious music of the medieval and Renaissance years that we find polyphony's roots. We might point to its genesis in the organa of those figures Leonin and Perotin, shadily enshrouded in the mists of antiquity, their work surviving only because no one felt it necessary to take those specific pages to recycle them as filler in book bindings, but it would take a hundred years or so for that polyphonic tradition to blossom into what we know as Mass composition.
Now, organum itself started out as a kind of polyphonic improvisation upon the various chants used throughout the Catholic liturgical year. The Magnus Liber Organi represents a point of maturation of the technique, when certain improvisations were given a level of "authenticity" -- we might think of it as the earliest conception of those works as "art" and worth repeating. But eventually medieval composers would take to setting the various chants of the Mass itself as complete cycles, instead of composing each part of the Mass Ordinary -- that is, the Kyrie, Gloria, Credo, Sanctus, Agnus Dei -- separately (the Ordinary being the part of the Mass which does not change through the year -- recall (for the first time, perhaps) that the liturgical year revolved around reading the Bible once through -- so that every Mass would have parts that were different, called the Proper).
At first, a composer might base each setting on a different chant, but eventually they began to use a single chant melody (generally taken from an appropriate chant whose sentiment reflects the day in the liturgical year, say, a chant celebrating a particular saint on that saint's day) as the basis or tenor of each part of the Ordinary, thus unifying the Mass quasi-thematically and modally. We refer to such masses as "tenor" or "cantus firmus" masses.
The cantus firmus mass came to be a popular way to write mass settings, but more techniques began to blossom as time went on and the Renaissance began. One is the paraphrase mass, which is a particularly English take on the cantus firmus mass. In these masses, the generating tenor is not necessarily stated faithfully in a single voice, as had been the tradition in cantus firmus masses. Instead, the melody of the chant -- or, increasingly, a tune of secular origin -- could be freely embellished or spread throughout the voices. Another technique is that of the "parody" mass.
At this point I might back up to point out that the development of secular song was also flourishing through the early Renaissance (in music), from about 1400 on. These secular songs -- such as the chanson -- took after sacred models in that they were polyphonic and were typically based on their own, secular "cantus firmi." A similar tradition of religious music ran parallel to the chanson, the motet, which is basically a piece of religious music that has no liturgical function besides to pad a service (not that the Catholic Mass of the fifteenth century had a need for padding -- such works might have been sung during normally silent parts of the Mass). The parody mass in turn takes these polyphonic works as models and follows them.
Typically a parody mass will follow its model by using the same music set to different text. So, as opposed to the paraphrase mass, whose model you find interspersed in a four- or five-part texture, you find the parody's influence in how the initial points of imitation, say, unfold, usually pretty exactly, so that the two works appear in their parts at first to be the same music save for different texts.
Not that exact imitation of the polyhponic models was always used. Indeed, a composer's virtuosity might be found in how the original materials are adjusted. A composer, for example, might postpone later entries of voices, thus changing the harmonic structure of the original work entirely in the new, or they might take out individual voices and insert new counterpoint. By the early sixteenth century, composers were having a heyday writing this kind of music for their chapels and patrons, borrowing from secular and sacred polyphonic works left and right.
It is with the early sixteenth century that the parody mass seems to decline. Not that it is subsumed by a new technique, but that mass composition increasingly becomes a conservative realm of composition, as opposed to the central part of a composer's work. Music printing really gets underway in the 1520s, and with it the popular madrigal comes to the forefront of musical innovation, leading us to the extended genesis of Monteverdi's "second practice," which would come to symbolize the beginning of the Baroque. While mass composition would continue, the parody technique itself implies a certain disregard for the meaning of the text it sets, which becomes the very antithesis of the second practice, and so it would come to be abandoned and eventually replaced by through-composition.