Help with squat form

You: overall not bad. Three things I see. One, I think you are barely hitting parallel, actually some reps look like you're not. If you want to get full value of the squat, try to go a bit lower. Second, are you trying to do high bar or low bar squats? The bar looks a bit low for high-bar. I'm not informed on how low-bar squats are supposed to line up vis-a-vis the bar path in relation to the foot, but for high-bar squats, the bar should fall in a vertical line pretty much over the center of your foot (imagine the middle of your foot being a fulcrum, where you line up the bar with that vertical path so you don't have those weights on either side of that vertical center. Third, you MAY be turning your knees in just a tad bit on the way up. Keep your knees in alignment with the angle of your foot.

Girlfriend: Wow, long legs!!! Again, overall not bad. Notice how her bar is more aligned vertically with the center of her foot (good!), and compare just that to your video to see more what I'm talking about (for you). Two things: work on going lower, she's not hitting parallel, and also just make sure (hard to really tell given the angle) that knees stay pushed out over the alignment of the angle of the foot.

In both cases, don't worry about knees going PAST the toe; that's been de-mythed in Rippetoe's Starting Strength. Just keep your knees aligned with the angle of your foot.

Also may want to get lifting shoes to take the cushion away from your heel, which raises your heel slightly and doesn't let your heel firmly push on the ground as you move. You want to use your heel and that force to push yourself up, and it's hard when you have cushioned shoes. Tennies shouldn't be used for squats.

Pull your knees out slightly as you descend. This will enable you to go lower. This is explained more here.

Your GF has poor depth. She almost looks afraid to go deep. Does she have an old back or knee injury?

I am absolutely no expert on squatting, but did read an article the other day that could help figure out how low you can squat and what your best stance is when taking hip mobility into account. Perhaps take a look at this and try the exercises in the vids.
http://bretcontreras.com/no-two-hips-are-the-same-how-anatomical-variance-can-affect-your-range-of-motion/

Ditch the running shoes, those are going to cause problems. Either squat in socks, barefoot or get a flat firm soled shoe (or true squat shoes).
Vid from the side plus the front or back.
Stance looks good. Might try turning out the toes a bit more and see if that makes a difference with depth.
I've had to rebuild squats a few times to correct issues, some I wasn't aware of until shooting vids from different angles. One of the best rebuilding methods I've used is doing a box squat. You can google it. Not only does it help with understanding depth you can drop the weight a bit and work on any weaker muscles that need strengthening.

Your body position looks good. To get a better video, I recommend recording from a fixed point around hip height- one video from the front, one from the side. It looks like your knees might be coming a little close together, but it's hard to say. Another angle would help. You look like you're maybe 1 inch away from being at parallel. My husband used to have the same problem. He thought he was nailing it. Then he had a friend show him how low he needed to go and he had to drop 75 lbs to be able to get to that point. If you have flatter shoes (or ones with less absorption, like chucks), you'll be able to spread your toes and grip better.

Your girlfriend is going maybe 75% to parallel. I recommend removing all weight and get the depth down first. She also looks like she's leaning forward to come up, but it's hard to say since the video is high up and not stable/flat. I used to squat just like that. It took dropping the weight completely to fix it. I also have to think "sit back sit back" every time I squat to make sure my hips are going far back instead of my knees going far forward. Leaning forward like she is now doing partial squats is more likely to injure her then doing full squats that go back adequately.

Much better but have your gf ditch the cross trainer/running shoes. They are not built for squats and can lead to issues down the road when weight gets added.
Keep a mental note of how your form feels in the second set of videos. If you pay attention to how your back stays straight, how your knees track outward and how your core stays tight it will help clue you in on issues when the weight gets heavier.

MUCH better.

A minor point - your right foot points out to far.

Is she wearing your shoes?

The biggest with both of you is your knees "caving in" or not tracking over the toes. You'd be surprised how fixing this, which is a glute activation issue, can fix other issues with your squat.