First, watch this video over on Starting Strength:
The Leg Press
In it Mark Rippetoe explains how to use the leg press to build up the strength required to complete a squat. I think Mark Rippetoe is engaging in a bit of hyperbole, here. The leg press is the only way? I just can't believe that.
The approach I've been using to take someone into a deeper squat is a bit different. It depends on a few variations - flexibility, strength, fear, and injuries.
If the problem is flexibility, in other words because they can't get to parallel in the squat without significant back flexion because of hip, hamstring, ankle, or other muscle/joint tightness, we address the flexibility. That means things like goblet squats with a hold at the bottom, squat-to-stand stretches, hamstring stretches, etc.
If the problem is strength, and it usually is, we work on improving their strength. Like Mark Rippetoe said, you can't strengthen a range of motion you can't get through. So partials don't work (not for this, anyway). What I'll do instead is try to strengthen the legs to parallel. My preference is to do this with step-ups, but I'll also mix in lunges, Bulgarian Split-Squats, and single-leg hamstring and glute exercises, too - especially one-legged deadlifts and glute bridges. But let's focus on step-ups. You can easily adjust a box or bench to put the working leg (the one on the box) to the exact bend it needs to be to be parallel in the squat. Yes, it's only one leg. But the leg takes a higher load than in a bodyweight squat. It's technically simple. It's easy to load (dumbbells, sandbags, barbells, kettlebells, chains, weighted vests) and it is easy to coach.
Nicely, you can do this even on people who can't load their spine. So if someone has a temporary back issue, you can do step-ups until it's healed up enough to load them with a squat.
If the problem is fear, we start with a box. I use all the stupid jokes about "sit back like it's a toilet" and teach them how to sit on chairs all over again. The box is often a bench. People often have a mental issue with sitting back and down. They're convinced they'll fall. With a box or bench behind them, one that they're sitting down to, they can't fall. Lower the box steadily while you work on other issues - the faster you get them in good form to parallel, the better.
You can also "weight" these with a tiny dumbbell, a ball, a empty water body, whatever, to ensure they have the right position without any added resistance making it more difficult. That will help them learn to squat with weight before they hold any weight.
If the problem is injury you have to either work around it or abandon the squat entirely. It depends on the injury, of course. For example, a spinal injury may mean squatting is out. A knee injury may mean you can't excessively load one leg. It's not always possible to squat.
These are my methods; I've also seen band squats and other trainers I know use ball squats against the wall. I suspect the progressive loading of band squats gets tricky, but it's another way to attack the problem.
Important Note: I'm not claiming Mark Rippetoe is wrong, here, not at all. I have nothing like the wealth of experience he has. I am sure he's absolutely right when he says the leg press will work quickly and effectively.
I just went right to thinking that, well, I don't have a 45-degree leg press and I've gotten people to a parallel squat in good form, even if I didn't subsequently back squat them. My methods may not be ideal, and there may be issues with them I don't see, but hey, I don't have a leg press. I have to make do.
I'd welcome any suggestions from trainers, trainees, or readers in general if you have any experience in this. If you have another way to get someone to weak to squat, to a good squat, without a leg press, please let me know!