Unless your alarm has chucked it, the parkers should be fed from the light switch and only get a positive "blink" signal from the alarm. So essentially they are separate circuits.
If your blinkers went off when you hit the alarm then sounds like the nominal voltage of the alarm was just enough to kill the last remaining juice in your battery. When you jump start, do the lights work? If not then you have a main line issue somewhere.
The immobilizer should only affect the running circuit - e.g. ignition or fuel kill. Other items such as lights and radio should still work regardless.
Be absolutely sure that when you jump start it you connect positive to positive and negative to negative. Reverse polarizing will damage the car's electronics.