Parking brake problems

Can't see the photo. Do I understand you have the parking brake inside the rotor separate from the caliper? It sounds like the parking brake cable is sticking. This is typical for a GM or any car or truck at this age and mileage. If it were a Ford, you would have replaced three cables already!

A leaking axle seal would be the cause of the grease. That can lead to a dragging parking brake shoe and the heat you noticed. First, have a helper work the parking brake and watch that the cable releases on its own. Replace the cable if you have to pry it back.

Caradiodoc

Nope, gear lube leaking is never a good thing. If you had drum rear brakes, the only proper repair would have been to replace the seal, the shoes, and the drum. The heat from normal braking causes the grease to soak into the linings and into the drum which is porous cast iron. No amount of washing or scrubbing will remove the grease to prevent it from leaching out every time it gets hot, and causing grabbing brakes.

Since this likely only affects the parking brake, I wouldn't be so concerned with cleaning it up, but the seal should be replaced. Most mechanics will recommend replacing the seal on both sides because the biggest expense comes from taking the differential apart. Once that's done, changing the second seal is just a matter of a few extra minutes and the cost of the second seal.

If you were to figure out that there is no unusual heat or smell from that brake after driving on the highway, but there is after doing some city driving, I would suspect the heat from normal braking is migrating in and burning off the grease. That would indicate no other repairs are needed other than the seal service. It still would be legitimate for the mechanic to recommend new parking brake shoes if they are very contaminated.

Caradiodoc