It turns out the parking brakes were a mess: the left cable was seized, some of the parking brake internals had come apart and gotten ground up inside, and as for the right side... the cable was cut and the hardware was missing.
Since we had the vehicle inspected at a dealer before purchasing (from an individual), and the inspection failed to mention that the parking brakes didn't work, the dealer agreed to fix it. A rough estimate of the cost comes to nearly a thousand dollars! And they even used a few non-OEM parts due to lacking availability.
Before driving away from the dealer we tested the parking brake... and it would not hold the vehicle! That was after the vehicle had stayed with them for a couple of days. So they the technician adjust it (again) asap while we waited. They said it may still need future adjustment as the cables and parking brakes wear in.
With the parking brake engaged and the vehicle in park, I can still get a slight movement by pushing the vehicle from the front or back. With the car in gear (drive or reverse) a slight tough of the accelerator will also get it moving.
How well do other Venza's parking brakes hold the vehicle?