Zadok the Roman seems to have found the first extant Catholic parody of "I Am the Very Model of A Modern Major General." It was written in 1951 by none other than Msgr. Ronald Knox, and runs thus:

I’m the sort of man they make an Apostolic Protonotary –
I’ve written reams and reams of prose, and quite a lot of poetry:
To walk on garden-rollers is among my minor glories,
And I used to be prevailed upon to write detective stories;
I can also punt canoes (or, as they say in Greenland) kayaks,
And had quite a flair at one time for composing elegiacs;
I can look up trains in Bradshaw*, on occasions locomotory,
As undoubtedly becomes an Apostolic Protonotary.

In short, when I’ve unravelled all the complicated mystery,
About what the Holy Office does, the Rota, the Consistory;
When I’ve studied more theology, and don’t get quite so drowsy on
Attending learned lectures which discuss the Homoousion;
When I’ve somehow put behind me (with my poor command of French) a list,
Of authors whose philosophy is known as Existentialist,
When my learning on a multitude of themes is less bucolic –
There’s ne’er a Protonotary will be so Apostolic.

(Notice the way all the parodists have to get 'Homoousion' in there somehow. The Whapster version has 'the substance Homoousios.')

Then comes the parody I turned up in Michael Novak's first novel, published in 1961 - The Tiber Was Silver. (Major teen angst at the North American College on the eve of Vatican II.) He has the seminarians sing,

I'll be the very model of a Doctor of Theology,
As positively certain as a Doctor of Biology.
I'll catalogue the Summa with new IBM machinery
And pop statistic certitudes like buttons in a deanery.
I'll study probabilities and probably pontificate
On what we'll find in outer space or on the moon (at any rate).
I don't think evolution's true but will not try amending it;
To make a monkey of myself has few points recommending it.
I've figured every problem out from books and texts and manuscripts
Until the year twelve thirty-one (when Thomas reached the age of six).
I've only seven centuries more to add to my anthology.
I am the very model of a Doctor of Theology!

These creative Roman seminarians also make up a parody of "Three Little Girls From School," which they sing while doing the can-can in their cassocks, with their birettas worn "rakishly at the back of their heads." Ummmmm... no comment. Overall, the book is an odd mixture of passive-aggressive rebellion against "pre-Vatican II rigidity," with unabashed triumphalism - Romanist aesthetics being so dang cool, how could anyone not gloat about them?

Then there is the Whapster version, which is the longest and uses the most outlandish words (extra points for putting in straight Latin and making it scan... sort of...)

I am the Very Model of a Modern Vicar-General

from the Penzance Codex of St. Gilbertus of Sullivan
translated by Matthew of the Holy Whapping and Lauren of Cnytr

I am the very model of a modern vicar-general,
I've information liturgical, ecclesial and clerical,
I quote the Popes of Latium and councils ecumenical,
From Chalcedon to Vatican, with subjects esoterical.

I'm very well aquainted too in matters sacramentical,
I know the sin occasions both the distant and proximical:
About the Nicene Credo, I'm teeming with a lot of views:

(Views...views...views...ahahaaa!)

With many complex facts about the substance Homoousios!

Chorus of Seminarians: With many complex facts about the substance Homoousios,
With many complex facts about the substance Homoousios,
With many complex facts about the substance Homoousi-ousios!

I've very good recessional, antiphonical canticles,
I know the secret names of all the Jesuit conventicles,
In short in matters liturgical, ecclesial and clerical,
I am the very model of a modern vicar-general!

Chorus of Seminarians: In short in matters liturgical, ecclesial and clerical
He is the very model of a modern vicar-general!

I know salvation history, King David's and the Sampson locks,
I answer hard sed contras, and own a pair of scarlet socks.
Respondeo dicendum every Vatican concilius,
All liturgics I can celebrate in Romanist basilicas.

I can tell undoubted Augustines from Bossuets and Zwinglians,
I know a Sarum Epiklesis and excommunicate the Arians,
Then I can hum the Sanctus if I've heard the mode ex nihilo,
And sing in tono recto Pax Domini cum spiritu tuo!

Chorus of Seminarians: And sing in tono recto Pax Domini cum spiritu tuo,
And sing in tono recto Pax Domini cum spiritu tuo,
And sing in tono recto Pax Domini cum spiritu tuo!

Then I can write encyclicals in a monastical scriptorium,
And pontificate the meaning of St. Paddy's grand loriculum,
In short in matters liturgical, ecclesial and clerical
I am the very model of a modern vicar-general!

Chorus of Seminarians: In short in matters liturgical, ecclesial and clerical,
He is the very model of a modern vicar-general!
In short in matters liturgical, ecclesial and clerical
He is the very model of a modern vicar-general!

If you want to utterly short-circuit your brain, read the Presbyterian version and Tom R's version, written from the perspective of some frightening chimera he calls "Catholorthodoxy." By the time the Unitarians get to it, you know it's time to quit.