why a resistor is added in parallel with a transistor base-emitter. e.g., a transistor is driven through a 10k at its base and between base and emitter another 10k is connected in parallel?
Collector saturation is a direct function of the base current. However, overdriving the base means that more space charge must be removed before the transistor can be turned off.
What is driving the base? If it's something that may be disconnected, the resistor might be there to make sure that the base doesn't float when the driving source is removed. This will ensure that the transistor is off.
More details on your circuit will result in better answers.
If I understand your description, the Base-Emitter resistor is used to create a voltage divider with the base resistor. This voltage divider will guarentee that the transistor is turned on hard.
Sounds like a PNP... the base-emitter resistor is to make sure the base is pulled high (transistor off) when nothing is driving it (the base driver is most likely open-collector).
I've seen this, and have done it in some industrial designs - frequently on transistors that buffer signals or control relays or solenoids. In an environment where arc, sparks, and high magnetics fields from nearby transformers are a norm, it insures the transistor is OFF unless it really does have base current. Also, if a module is disconnected, the wire/trace to the base is less likely to pick up enough stray EM to momentarily turn the transistor on.