No pressure in one side after replacing rear brake calipers

A number of things came to mind. First, replacing just the piston is the worst way to go with this design. It has to be filled with brake fluid before it is installed, otherwise the air will never come out. Second, it is only replaced if the chrome plating has lifted and there's rust forming under it. Otherwise, it's the square-cut seal that leaks and must be replaced. Those come in a kit with one seal and one dust boot for one caliper rebuild.

The point of value is with this design, the pistons do not self-adjust by just pumping the brake pedal, like the front ones do. These are adjusted by applying the parking brake a few times. It's common on Ford products to find the parking brake cables rusted tight after a couple of years. In that case, you can use a large Channel Lock pliers to work the parking brake lever on the caliper by hand. Once adjusted out, the caliper will self-adjust for normal pad wear like normal on most models. On some, they only adjust when using the parking brake regularly. Until that caliper is adjusted, the brake pedal will be real low. The master cylinder's piston for one front brake and the opposite rear brake has to push brake fluid to those wheels, and it pushes fluid that pushes on the piston that moves fluid to the other two wheels. That required volume goes way up when a rear caliper isn't adjusted.

Next, the loading of minivans and pickup trucks can vary a lot. To address that and maintain balanced braking front-to-rear, most use a height-sensing proportioning valve at the rear axle. If the vehicle is supported on jack stands or on a hoist with the rear axle hanging down, it will look to that valve as though the rear is lightly-loaded, and less brake fluid pressure must go to the rear, to prevent rear-wheel lock-up. It is often necessary to set the vehicle down on the tires to open up the proportioning valve and get fluid flowing faster.

One of the things we see too often is after pedal-bleeding with a helper, the master cylinder is damaged or crud is pushed down one of the steel lines. By about a year, crud and corrosion build up in the lower halves of the bores where the pistons don't normally travel in the master cylinder. It's important when pedal-bleeding to never push the brake pedal more than half way to the floor to avoid running the rubber lip seals over that crud, and ripping them. Ford products often have four steel lines leaving the master cylinder instead of just two, so it is possible to have a problem with just one brake.

Solved!

Correction: I replaced the entire caliper assembly. Not just the piston.

To address some points by CARADIODOC:
The parking brake cable and whole assemble appear to be in good condition with no rust. Adjusting that is part of the solution.
This model has two lines from the master cylinder to the ABS.
There is no proportioning valve on this vehicle.
I didn't know about the crud problem. I'll keep that in mind next time I pedal-bleed a system.
Thanks for the input Cardiodoc.

Solution:
After some though (mostly by a knowledgeable friend) I decided to change the front brake pads and try again. Why? The thought is that the master cylinder ran out of play. That is, it was expected to push too much fluid to engage the brakes. Replacing the front pads reduced the distance the cylinder had to travel.
Additionally, I believe that by playing with the hand brake we adjusted some of the play out of the rear calipers.
Finally, I have four wheel disk brakes, with the bleeder valve at the top of the caliper we are able to simple open the valve and let fluid drain out.
After all this the brakes feel like they did before I started this job.

Dandy. The bleeding method you described is "gravity-bleeding". That is all I've used for the last 35 years. Back in the mid '80s, I saw an experienced mechanic punch the brake pedal all the way to the floor on a Ford van, and it stayed there. It even pushed mud into the combination valve. A simple brake job with rebuilding the front calipers turned into a two-day nightmare.

You're right about solving the low pedal by adjusting the parking brakes, but that doesn't apply to the front brakes. The feature of all disc brakes is as the pads wear, the pistons work their way out of the housings over time. That is how they self-adjust. If they did not do that, every car and truck on the road would have a brake pedal that went all the way to the floor every few thousand miles. Replacing the front pads was not necessary. When you replace them, the old ones have real thin linings, but the brake pedal is still up where it should be.

The only time the front calipers aren't in adjustment is after the pistons were pried back in to make room for the new pads. The brake pedal has to be pressed, usually multiple times, to run the pistons back out until the pads contact the rotors. That is when pressure in the pedal will build up. This is where a lot of do-it-yourselfers and inexperienced mechanics damage the master cylinder. They run the pedal all the way to the floor, and that can rip the lip seals. I always pretend there is a block of wood under the pedal.

Thank you for posting the solution. Happy to hear you solved it.