From the sound of your problem, it’s a safe bet that something has caused your transmission to overrev and the transmission control module — the electronic brain of the tranny — has put the device into a serious form of limp-home mode (LHM).
Normally, if the transmission control module determines that an overspeed condition has occurred, the transmission is locked into one forward gear, second, and reverse. What can cause this type of limp-home? It is quite likely that there has been some minor gear slippage that has resulted in the overspeed. The slippage could be the result of a momentary torque converter glitch, or, possibly, an improperly activated servo or solenoid. In each case, unique codes are generated and stored and the transmission goes into “standard” LHM (second/reverse).
Since momentary glitches are considered minor by the transmission control module, they can be temporarily cleared by taking the action you described — foot on the brake/park and cycling the key on and off. It usually takes two or three cycles to restore semi-normal transmission operation. Notice that this is semi-normal operation. In some cases, if the glitches are small, the transmission control module will not store any codes and will then allow the reset to become permanent. This isn’t the generally accepted action of the control module. Instead, it does allow the temporary reset so you can drive more normally, however, once you reach your destination LHM takes over again.
If the slippage is more severe, it is quite possible that the transmission control module can set more severe LHM limits, placing the Sprinter into third gear and reverse. In some cases, the slippage, combined with overrevving, creates the condition that you have reported. Your Sprinter is actually defaulting to a new gear. In this case, instead of second or third, it is defaulting to neutral and your transmission is disabled. If this is the case, it is unlikely you will be able to clear any generated codes with anything other than a diagnostic reader, so the transmission will stay disabled until you get to a dealer or specialist.