It sounds like; in the process of removing the bumper, you disconnected the ground wires or disturbed the ground connection for the rear light assemblies. The electricity that runs the lights comes through the harness to the bulbs; but it doesn't stop there. After lighting the bulbs, it then has to return back to the battery to complete the circuit. The return path for the electricity normally flows from the bulb housing through a ground wire which is connected to either a body panel or the vehicle frame. The power then flows through the body panels or the frame, until it reaches the battery ground cable, under the hood.
Since the engine is mounted on rubber mounts, and the main battery ground cable is normally connected to the engine; there usually is a poor electrical contact between the engine and the body or frame. So, in order for the vehicle's electrical system to work properly; it is necessary to have an additional ground cable connected between the battery negative (ground) terminal and the body; and often also between the engine and the firewall, so that electricity used in other parts of the vehicle can return easily to the battery. If both of these auxiliary ground cables are removed; it can create all sorts of weird electrical problems.
Similarly; if the ground cable between the taillight assemblies and the body is disconnected, all the power that is used in the taillights then tries to find another return path to the battery. The only alternate path that is usually available is between the mounting screws for the taillight housings and the body panel they thread into. These screws and panels are often rusty or corroded; which makes a poor electrical connection. As a result; there may be barely enough contact at that point to allow the amount of power used by a single bulb filament (either brake or taillight) to return to the battery; but when two bulb filaments are lit at the same time, it puts more power through the ground circuit than the connection can pass. When that happens, since the resistance of the connection at the screws blocks any more current than the amount used by one bulb filament; when the second filament starts receiving power, the whole circuit overloads and both bulbs go out. The way to fix this is to run a new ground wire from the light assemblies to a proven good ground point (however far away that may be). Since you say that the rear light on top of the cab works properly, you probably can use a clean bolt (with the paint on the panel scraped down to bare metal) in the body panels of the cab as a ground wire. It is not uncommon for the bed of a pickup to become electrically insulated from the cab.
On vehicles which have become badly rusted from operating in salty environments; the rusted welds between adjacent body panels will sometimes develop enough resistance to block the flow of electrical power. When this happens; the rear lights will develop problems. One way to fix that kind of problem is either to run a ground wire from the light assemblies to a clean bolt that is threaded into the frame, or.on unibody vehicles which do not have a frame; to run a 12 gauge stranded electrical ground wire from the light housings all the way to the battery ground cable.