: Rules for use of quotation marks when paraphrasing quotes, song lyrics etc., for a humorus effect in a cartoon caption. When a person/writer paraphrases published quotations or song lyrics, changing

Some references to popular songs would count as "fair use". For example, Sir Terry Pratchett, in his novel Soul Music (which is all about Rock) makes references to multiple Rock songs.

For example, there's a song called "Sioni Bod Da", which translates from Welsh as "Johnny be good", and references Chuck Berry's song "Johnny B. Goode". The character who sings it also has the line "I can feel it. Every day. It's getting closer...", referencing Buddy Holly's song Everyday: "Everyday, it's a-gettin' closer / Goin' faster than a roller coaster". (You can see a full list of references here.)

In no place does Pratchett give a list of "those are the songs I referenced" - it's for readers to piece together the references, and have a good laugh when they recognise them. (Or stare blankly when a reference goes over their heads.)

Making hidden references to real-world stuff is the fuel much of the Discworld runs on - it's what Terry Pratchett does. It's humorous, and the quotes are never extended (in fact often there's no exact quote at all).

That said, I would recommend checking with a legal practitioner what is covered by "fair use", and what isn't.

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Simply putting the material in quotation marks will not avoid plagiarism. Further, in your specific example, putting a paraphrased version in quotation marks—without any commentary—would be incorrect. (Except in the case of character dialogue, which I discuss later on.)
To avoid plagiarism, you need to provide a proper citation for the material (ascribe credit to its author). You also need to either provide the material verbatim (within quotation marks or a block quote) or you need to to indicate that you've made a change in some way.
For example, from MetroLyrics, here are the lyrics to the first three lines of the first verse of the Beatles song "Let It Be," written by Paul McCartney and John Lennon:

When I find myself in times of trouble
Mother Mary comes to me
Speaking words of wisdom, let it be

Note that I have provided the name of the song, its authors, as well as the specific website from which I copied the lyrics. Different websites provide different formatting for these lyrics. For instance, Genius puts quotation marks around the occurrence of let it be in the above passage. So, it's not only important to cite something's original source but the source of the material you've actually quoted.
Let's say that MetroLyrics made a typo on their website-for some reason, reversing the order of times of trouble. I would quote it with the error but also call it out:

When I find myself in trouble of times [sic]
Mother Mary comes to me
Speaking words of wisdom, let it be

Here, the use of [sic] is the traditional indicator that there is a typo in the source material, rather than in your quotation of it. But since I already mentioned the error, I didn't need to actually provide it. (I only did so as an example.)
Also, different style guides will give different guidance as to how to handle this. Some will say that small errors can be "silently" corrected, others that all instances should be noted in some manner. And whether the reversal of the word order here would be considered a "small error" or not would be up to the particular style guide.

But let's say I'm paraphrasing this. I can't just do the following.

Here are some lyrics to "Let It Be":

Sometimes when I'm in a spot of bother
My mom Mary pays me a visit
Advising me to just forget about it.

Even if I had given an appropriate citation for this (more than just the title of the song) I need to make it explicit that the words in the block quote are not the original words of the song.
A better way of doing this is to be more explicit.

Imagine if you will that Paul McCartney and John Lennon had written different lyrics to the Beatles song "Let It Be":

Sometimes when I'm in a spot of bother
My mom Mary pays me a visit
Advising me to just forget about it.

Here, I've given appropriate citation to the original source. I've also not mentioned any particular website I got the original version from because I'm changing it so much anyway. Last, I've made it explicit that what I'm providing is something different from the original.

Now, normally when you paraphrase something, you do so in simple terms and without quotation marks or a block quote. You still need to provide some level of reference, but it doesn't need to be as great.

Imagine if Paul McCartney had had a dream about his father visiting him rather than his mother. Maybe he would have ended up writing a song about worrying about your problems rather than just letting them be.

Here, I'm providing just enough information that most people will understand what I'm referring to. But the specific content from the song lyrics is so minimal that my commentary doesn't need to be in any kind of quotation marks or other formatting. Nor do I need to do more than just mention Paul McCartney—because I haven't actually quoted anything.

A final way of doing something along the lines you're talking about is to, for instance, have a character in a bar sing the lyrics to a song—but sing the wrong lyrics. In the actual story's dialogue, that's fine. You don't need to necessarily mention what song it is, who wrote it, or what the original lyrics are.
You can have the character just sing the wrong lyrics and carry on with the story.
But I've often found that when a story does reference a song in such a way, credit is given to the song in some fashion in the front matter or the back matter. Since it's being used in character dialogue, the standard rules of attribution against plagiarism don't apply.
This may be similar to how it would be handled in cartoons where a cartoon character is quoting something in dialogue.
However, I would rely on a more authoritative source than just what I've said here if you want to incorporate something in such a specific fashion.

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