An example would be green beans - some recipes call for parboiling green beans before stir-frying while others do not? Is there a reason behind this?
Really, this is just to even out cooking times for vegetables that don't have a surface area to volume ratio consistent with the other things you're stir-frying. If you were to shred those green beans, as is sometimes done, you could put them in at the same time as raw, julienned carrots, and they would finish at the same time.
If you put them in whole, you'll end up with beans that are burned on the outside and raw on the inside, as there's not enough time for the heat to penetrate to the center of the bean before the outside burns. Or, if you fry at a lower heat, you could get the beans cooked, but would have to add the carrots later (which means a lower temperature, too, since the beans are dragging heat out of your pan) in order to have them not turn to mush.
I'm a convenience stir-fryer and won't bother with parboiling. Instead, just use the staggered ingredient method. Green beans do take longer than ingredients like, say, shrimp. So line up all of your ingredients in order of cooking time and stir-fry accordingly. It takes some experience, but onions/garlic then dense starches (potato) then fibrous vegetables (broccoli) then proteins (chicken) then delicates (fresh herbs, lighter greens) is a good framework. I use this method all the time and don't have to bother with multiple preparations.
note: A variation/alternative to this method is to start with your protein, brown it nicely, and then take it out of the heat to be returned later. Especially good with proteins that can easily overcook, like shrimp.
Whilst sometimes you need to do this to cook vegetables which are 'thick' or require a fair bit of cooking to make them edible (beans if they are whole, broccoli unless it is broken up very small), what I prefer to do is to stir fry these bits together (along with some garlic/ginger/chili if i'm using it) at the start and when they have taken on a little colour (a minute or so) I add a few tables spoons of water, turn the heat down and put a lid on the wok.
This allows the veggies to steam and so be sufficiently cooked. Then I take them out and keep separate, cook the rest of my stir fry and add them back in before I add the sauce.
I always blanch before stir frying, which is slightly different than parboiling. Parboiling is boiling for a set amount of time, generally (for green beans) 5-7 minutes. Blanching is faster, 1-2 minutes max at my house, and then dumping into an ice bath to stop the cooking process temporarily.
The benefits I've found are:
1) More even cooking. By starting the beans in boiling water, as Rich said, it allows the centers to start to get cooked whereas with just stir frying I've had raw centers before. When I sauté raw green beans, the result is generally beans that taste very "earthy" and a lot more raw - depending what you're going for, this can be good or bad. My wife doesn't like the taste of raw vegetables, but does like the crisp texture, so by blanching I'm able to begin cooking while still maintain their crispness, and also make them much more palatable to her. When I parboil, I tend to end up with slightly softer/mushier beans.
2) Maintains the bright color, sometimes even enhancing it, very important to me when serving guests as you eat with your eyes first.
3) Enhanced/locks in fresh flavor. I don't understand the science behind this yet, but experience and trial and error have shown me a significant difference between when I do blanch, and when I don't.
I know that 1) will apply to parboiling, and is the main reason you would parboil before stir frying, but I'm not sure if 2) and 3) would as well.