Do try things - unless you know you should not.
Here I give general pieces of parodying advice, to give an overview of the important considerations of parodying.
The set of characters I ultimately settled on using for my parodies, the Black Pearl Crew, are my standard reference for the kinds of things a parodist should do.
To a non-parodist, this may sound like an odd suggestion, since parodies, by definition, are not serious. The considerations that go into making them good, however, can be quite serious. Parodying songs is much more complicated than it looks. Were it not such a rare occupation, entire books on how to do it would probably be written. It is not something you should take lightly, unless you want to be the subject of stories like these.
Basic pop and rock are the best places to start, but you will not want to stick with them forever. Songs that are more complicated or cover unusual topics can allow you to do things that do not work well with ordinary songs, and more difficult songs are more gratifying to make good parodies of - if you can manage that. However, there is a point where unconventional songs stop being an opportunity and start being an impediment. Before accepting a challenge, make sure you can see that you have something to gain from doing so.
Possibly the single most important thing a parodist has to realize is that you cannot just make any song about anything. Oh, sure, technically you can, but make sure you can sound reasonably close to the original words. If you want to tell a tragic love story set in the 1600s, do not pick a song about 21st century American politics, religion, et cetera. If you try to put a story into the wrong song, you will end up with parodies like these. And some stories are just not a good match for any song, or vice versa.
Trial and error is unavoidable as you learn to do something, so you might as well make it your goal. It is best to start out wide open and develop rules as you go, once you have experience with what actually works for you and what does not. Also remember that your rules are your own, and you can change them any time you want.
As I just said, trial and error happens naturally, whether you mean it to or not, and is therefore worth embracing. For as long as the Crew have been established as my parodying characters, I have been doing things in my parodies I had not done before (see here). And then, in 2019, I took a turn that would have been unthinkable six years earlier!
It is essential for everyone to know their limits. For example, being very prolific can be impressive, but only so long as it does not reduce you to doing a rushed job. I went into rapid fire-mode upon realizing I could write a whole parody album at once, writing the first three albums I did as such in just as many months. But then I went two months without parodying, so I must have realized that assembly line parodying was not necessary.
Likewise, by 2018, I had a general feeling that my parodies had already had their height, and the numbers of them I have been doing have dropped accordingly. I have long known what I want my parodies to be. I have used most of the best material I already know. I have little, if anything, left to experiment with. I can still have fun writing an occasional new parody, but nowadays I am fairly content to bask in what I have already done. Having parodied more songs than many artists record in their entire careers, I have been plenty impressive as it is.
One part of all that of particular note is that by using Kilroy Was Here, Paradise Theater, and American Idiot the final time each, I entered great challenges and came out with excellent results like never before - or again. And the important thing is, I knew it. I knew I had chosen the right concept albums. I could have continued the trend with 21st Century Breakdown, but I wisely realized that album is just difficult in a purely difficult way and would not have rewarded me for taking it on the way its ilk did.
Before the crew of the Black Pearl, I wrote my parodies from the perspective of some of their Lego video game counterparts, whom I dubbed the Comic Five. I stopped using them for a reason or two - or fifteen. They are now my standard reference for what a parodist should not do.
Every entertainer has to write things undesirable to publish as practice before they can create things worth releasing, and I was no exception. You can find that out for yourself here. It is inevitable that you will have to go through a rough phase like that yourself - but at least I have been through trial and error so you will not have to. I hope my advice can make it so you do not have to reinvent the wheel like I did.
The Comic Five were notorious for, among other things, doing everything they did because they thought they had to, not because they wanted to, especially because they had some very strange ideas of what was necessary. As you can see from reading their discography, linked to above, nothing reduces the quality of a creation like a maker who does not care.
The Crew decides what to sing about and when to sing about it based entirely on what the song has to offer. When it comes to parodying, everything needs a reason, which is why I have that practice.
The song and the parody subject have to go together in some way. You need a better reason to be doing this song than just that it is the next one in its album, and a better reason to choose this parody subject than just that it is what was going on lately.
I do realize this probably sounds like a contradiction of my earlier advice to discriminate, but the aforementioned Comic Five were a good example of how being picky and indiscriminate at the same time is possible. They only wanted to parody songs that were in my 4-5 star rating range and that my readers had already known for years - but they would try anything that fit those criteria, no matter what it was. While it is true that the parodist needs to like the song well enough and knowing your audience helps, if you approve a song to go and stay on your iPod yet insist that you do not like it enough to parody it, or decide you cannot parody a song because you know someone who does not know it, you have gone too far.
I learned just how unfussy I should be with Green Day going solo acoustic guitar in Ordinary World. Normally I would not even have such a song on my iPod, but then my friend pointed out that it mentioned buried treasure. Ultimately I dropped any complaint, having known perfectly well since two years before the song even came out that such instrumentation only enhances the piratical setting, anyway. (That was, in fact, one of the catalysts in the Crew taking over from the Five.)
Then came the Rick Springfield album Tao. I never used to listen to this one song off of it, The Power Of Love (The Tao Of Love), for its being a repetitive slow song. But nothing will ever have a chance if you do not give it one. I realized I should at least check to make sure it is not a 5-fodder song.
It was.
Ambition is admirable, but remember, technically, anyone can parody any song. The question is how good a job you can do. If you want attention, you need to do more than just get noticed - you have to be worthwhile for the viewer, too.
As I noted above about when to relent, I purposefully do not get all Look at me, look at me writing more parodies every month!
or Look! Look what a hard album I am doing!
I get prolific or choose challenging songs as appropriate, never for the sake of it, knowing the latter does not actually give me any bragging rights.
I have previously advised you take parodying seriously, but there are limits to everything. In short, your life does not depend on your parodying, so do not do it as if it does.