If you hang around me long enough, you will probably see me smite something with my fist and hear me thunder (inasmuch as altos of my size can thunder), "Parallel structure: it's a privilege, not a right!" Most recently there were a few Wonderfalls episodes that provoked this response. (Wonderfalls: not my show. Really not.) But it comes up a lot. And it's not that I hate parallel structure. Some lovely things can be done with parallel structure--see, for example, The Wire. But I think it gets overused, and I think it's the first thing people pull out when they want to do a trick with structure, and let's think about it and maybe put some more structure tricks in the bag, okay, kids? Yes. Okay. Here we go.
Counterpoint/perpendicular. Okay, where parallel structure is having the same thematic thing happen to more than one person, this one is illuminating theme by having the opposite thing happen around the same idea. The central arc is doing this. This subplot/sub-arc is going the other way instead. While your main character is falling in love, their best friend is falling out of it. While your protagonist is finding their life path, their mentor is filled with doubt about theirs and genuinely leaves it. (A doubting mentor? Will you do that for me, please? How many mentors actually say, "Um, you know what, kid? This heroing business: not so great actually. Let's go off and open a tea shop in the Boulevard Saint Jacques. Splendid. You make the pastries and I will buy the teas and talk to the customers about them. Good. What, what do you mean there is a giant magical beastie crashing against the door? Damn. All right, magical beastie first, tea shop second.")
Spiral. Do the thing in very brief form. Do it again a little larger. And then again a little larger still. And then larger than that. How many times this happens depends on how large your end product is; Greer Gilman wound up with a whole novel that way. (Are you Greer Gilman? The odds are against it rather overwhelmingly--several billion to one, is I believe the current count. But that doesn't mean spiral structure is beyond you; most of what makes Greer difficult for the people who do find her work difficult is that they don't have the background to find her language and reference clear, not that her structure is difficult. I think you could borrow Greer's ideas on structure and be a great deal more commercial than her work is, if you want to. Or, y'know, not; certainly not everybody has to be aiming for commercial.) The other example close to the top of my head that is not Greer is that a lot of symphonic work uses spiral structure: introduce the theme in very brief, develop it, develop it at greater length, come thundering back to it to develop at yet greater length, and like that.
Cascade. Is like parallel, but at radically different points in the arc. Hard to pull off without lapsing into parallel, though.
Aspect/list. Divide the work into sections. Does not have to be a classical number of sections with classical labels--probably should not be, for maximum interest. Go for the weird divisions. The stranger aspects the better. skzbrust did the laundry list and the meal. Those are good, but he did them; if you do them, it'll be "oh, cheap attempt at Dragaera homage."
Trick riding. If I can come up with these as general categories, you can look at your actual work and see something I can't see because I'm not looking at your actual work. Right? Maybe? It's worth a shot, anyway. And if not, maybe a structure trick is not what your work needs.
(I think they're particularly common in TV because TV has a set time length, so if you have a plot that doesn't quite fill that, you want to do something with the secondary characters, and you want it to look not quite random and tacked on, so...structure trick! Often sitcoms do not even bother with this. This becomes particularly transparent when you read their summaries. "Vanessa likes a boy. Meanwhile, Theo gets a bad grade." Really? This was the best we could do, America? Really? Sigh.)
Any other structure ideas for things that don't have to go in parallel?