I cannot select bones once in weight paint mode. I have already turned off "Lock object modes" and I can switch between pose mode and weight paint mode so that's not the issue. I can go into pose mode and move my mesh around so I know it's parented and I know it has an armature modifier. I am not sure what I am doing wrong. Here is a link to download the file. I am using the latest version of Blender at the time of writing this 2.82
In 2.91 this is no longer working. Selecting a bone in weight paint mode throws both the mesh and the armature into object mode. Then we have to switch back to pose mode in the armature, then back to object mode, then select the mesh, then switch it to weight paint mode.
I did have vertex selection off, paint mask off, and still can't select bones in weight paint mode using ctl-left click in v2.91. Or any of the many methods I've tried. Having to switch from pose mode to object mode, unable to shift select armature to mesh when the armature is in pose mode, all very inconvenient.
Weirdly, when I was experimenting with temporary solutions to the Second Life/Open Sim armature (yes, a horror of an armature, but that's what I got hired to do) Blender 2.91 weight painting got less buggy when I conformed the Left Right bone names and fixed the hierarchy to list left bone bone chains first then right bone chains. Symmetry in weight painting started working better and I was able to ctl left click a bone in weight paint mode - not able to rotate it, but to get back to pose mode with fewer clicks.