Brake Caliper Piston will not compress?

Hello, you have the dust seal removed from the caliper piston I assume, and the piston is fully extended out. was that side set of brake pads more worn down than the rest? If you can't get the piston to rotate in even with the bleeder screw out, then the piston is frozen in that caliper. There is some rubber grease that is to be applied under the dust seal for the piston, but I think that piston is seized, and the caliper will need to be replaced. You don't want to take a chance with that piston not applying a braking to that particular caliper. The brake pads will wear out on the opposite side much faster, and you will have uneven braking power as well. I'll post the diagram about the grease for the dust seal. But you're safer just replacing the caliper.
What size engine do you have?

By chance, the Parking Brake lever on the caliper is not pulled in, is it? Sometimes the cables can get dirt or rust build up and fail to release the parking brake, The vehicle diagrams I am looking at are for a 2.5 Liter engine, although I believe the 2.0 is the same set up.

https://www.2carpros.com/articles/brake-caliper-replacement

I revised my post above, is the Parking Brake lever on the caliper pulled in? Sometimes the cable will bind, or dirt and rust can cause them not to release fully. Since the parking brake is part of the caliper on this vehicle. Do you have the cable off the caliper and if so, does the lever move freely on the caliper being disconnected from the cable?

You can see on the back of the caliper where the cable will pull on the lever, this causes the piston to be drawn out, engaging the Parking brake. If the piston is fully extended you should be able to move the lever back and forth, I just want to be sure it's not the parking brake cable that was holding things up. But the piston is not serviceable. You're better off just replacing the caliper to avoid any future issues. If it's seized into the caliper, it's not going to spin back in. And if it did, with an impact gun or by force, it's just going to seize back up and prevent braking on the tire.

Try taking the cable off and see if the piston moves then.

You should be able to pull the lever forward. Just like the cable would be doing if the Parking brake was applied, then you will have some slack in the cable to remove it from the caliper and see if the piston moves. Yes, you are pointing to the parking brake cable. You will have to use a pair of pliers to pull the lever forward because of the large spring.
But I do see a lot of rust on that piston, so it might be seized.

So, if the piston is seized into the caliper, the caliper will have to be replaced. Once you replace the caliper, reinstall the cable on the new caliper, bleed to the caliper and then go through the parking brake lever inspection process. I'm not sure why you pulled the Parking brake handle with the cable unhooked. The spring on the caliper is what pulls the cable back after releasing the parking brake handle.

No, the whole caliper needs to be replaced. I can see all the rust on the piston in your picture. It's seized up. With the piston that far extended you won't even be able to get the pads on. Make sure you ask for the correct side, on that caliper it has a LH, so make sure they give you the correct driver side caliper.

Only if you have an issue with the other side, if there is rust on the piston while extended it can be cleaned off. The rubber seal that goes around the piston should be changed if you can get them. The new caliper will come with one, and service info states to put some rubber grease under the seal to prevent dirt and rust from developing on the piston. I'll see if I can find out which type, they recommend. If you feel uncertain about the other side and there's rust build up, and can afford to buy a 2nd caliper, go ahead. You just don't want to rotate that piston in with any rust build up on it. That can cause it to stick in the future. If you have uneven pad wear from side to side on either the front or rear calipers, that indicates one piston is not moving as much as the other side.
Here are a couple of guides to help. And I'll see if I can find that type of rubber grease as in the diagrams.
One other thing, with the piston that won't move, have you tried pushing it in and rotating at the same time? Sometimes due to that rust, they get stuck on the outer edge. If you can clean some of the rust off, you might be able to get it to move. I know they can be difficult after so many years, especially if you live in an area where they use a lot of salt on the roads during winter.
They do sell these seal kits with a little packet of the grease, but looking at your picture again, was the seal missing on the piston when you took it apart?

https://www.2carpros.com/articles/brake-caliper-replacement

https://www.2carpros.com/articles/how-to-replace-rear-brake-pads-and-rotors

Thats really rusted in there. Spray that with some penetrating oil and see if it will loosen up. You must live in a place where they use salt on the roads during winter. I'll see if I can find some info to help you, I know these are not the easiest to do.
Here is a great video to help, but you'll have to let that cable soak in some penetrating oil for a bit, but it will come out eventually.

https://www.youtube.com/watch?v=8MZYpcYyWE4

In the video she is using a brass punch because it will not cause a spark and penetrating oil is flammable. Just use the same technique that she does in the video, it will come out. And congrats on the new baby, I hope all goes well for you.