Since people love to contrast Batman and Superman so much (I could rant on that someday) in my view that parallelism/contrast doesn't entirely work without contrasting Luthor and The Joker also. Because each of them is like every good Arch-nemesis should be the Antithesis to to the Heroes they oppose.
Batman embodies Order while The Joker embodies Chaos. Superman embodies Selflessness while Lex Luthor embodies Selfishness. There was a time when these contrasts were defined more superficially. Superman is Brawn while Luthor is Brains. Batman is good disguised as something scary, while Joker is evil disguised as something that delights children. But nowadays no one wants Superman to be stupid and few want Lex to be a weakling (which is the complaint about Eisenberg not being Intimidating, but I'm all for it). And Clowns are now universally considered scary in any context.
So if Batman and Superman are somewhat opposite kinds of heroes, and Joker and Luthor are their total opposites. Logic dictates they should be in some way opposite kinds of villains.
I prefer to define Superman as the Rebel Seigel and Shuster wrote rather then the stupid Frank Miller perception (Chaotic Good rather then Lawful Good). The Joker obviously is Chaotic Evil, he's become the definition of it.
Lex Luthor I have always felt works as an evil counter part to Batman better then any of the failed Batman characters created for that intent. Both are about Order in a sense. Batman has the best rouges gallery over all, but not a good Evil Counterpart, The Joker is a perfect antithesis, and many others have some parallels. But the ones made to be Batman if he were Evil (like Zoom is for Flash and Sinestro is for Hal, and the Phantom Zone villains for Superman) tend to flop.
I read this article on The Joker
And I hate it. The Joker is not about control, that is Lex Luthor. The Joker is about making other people lose control (Batman first and foremost but he will target others from time to time). That's his blatant objective in both The Killing Joke and The Dark Knight. Seeing someone who says they've read TKJ so monumentally misunderstanding it really annoys me.
Also The Joker is Asexual in my opinion (perhaps Homoromantic in relation to Batman) but completely without any sex drive. So no he did not rape or sexually assault Barbara Gordon in TKJ.
Lex Luthor however is a very sexual Alpha-Male, the early CEO Lex who was very unattractive was the embodiment of why we have sexual harassment laws, when he later morphs into being more attractive he's more of a womanizer, which is an unfortunate implication since attractive people can be rapists too. And slash shipping Lex and Clark on Smallville works very well.
The Batman villain who is a Sadist is Crane/The Scarecrow, but he gets off on psychological torture rather then physical torture, which is what Cillian Murphy captured perfectly in Batman Begins. His parallel to Batman is that both use Fear as a weapon, one for good and one for evil.
Writers haven't put the thought into how to contrast Luthor and The Joker they have into Batman and Superman. Many Comic stories using both wind up seeming randomly lame. However the World's Finest story-line on Superman The Animated Series did a decent Job. I also enjoy their moment in Infinite Crisis, and I rather like what The Joker Blogs is doing.
What about the Pre-Crisis Lex? Isn't he less Lawful and more Chaotic? Maybe, but he's still about power and control, in that era his methods may have seemed more similar to some incarnations of The Joker, but the goals were still different. As far as doing in the Pre-Crisis Lex in a more developed way I enjoyed the Allstar Superman animated movie. How he was in official continuity between losing the Presidency and Infinite Crisis also worked well.
Here is one interesting article on them that I mostly agree with
One finale important difference, that the article I just linked to didn't get. The Joker has wanted to kill Batman in some stories but I prefer The Joker to never want Batman dead, he's more interested in making Batman break then kill him. While Lex may want Superman alive at some times but ultimately his goal to be the one who kills Superman, which is why he was PISSED OFF at Doomsday.