Can we operate a 6VDC Relay using Parallel Port signal, without a Darlington Transistor, with only these components?
It's likely that your parallel port does not provide a high enough voltage. You can check this by connecting the relay to a variable power supply, start with ~4V and increase the voltage until your relay clicks. If the voltage on your power supply needed is higher than that from your parallel port, it does not give you a high enough voltage.

If your parallel port however provides enough voltage, the problem most likely is current. The parallel port will not provide enough current to switch the relay, and you're most likely damaging your computer.

So the answer to your question is no.

Anyway, for anyone looking for a working solution: it will work with a transistor like this circuit:



This uses a 50 Ohm resistor because the relay is 6V / 100 Ohm.