Well, it would seem to depend in part on the model and the year of your Kubota.
The earlier and least expensive B models (i.e. B7100, B6100, B5100, B9200, B8200, B7200, B6200, B1550, B1550, B1750, B2150 ... at least those tractors that actually had an hour meter) had an electric meter that started to count as soon as the key was turned on. These meters did not require the motor to be running in order to advance.
In my limited experience, the same is true for all of the earlier BX models including the BX22 and BX23. However, that changed around 2006 with the introduction of the multifunction LCD dash panels. For example, on the newer models with the LCD hour meters, such as the BX24, the engine must now be running in order for the hour meter to advance.