The Outlaw trigger outputs are typically spec'd at 12 volts, 50 mA (I couldn't find the 1050 manual on line but the 1070 and 970 agree on the 50 mA). The trigger input on the Outlaw amps are spec'd at 5 mA minimum drive current. I couldn't find anything on the Parasound requirement. But unless it's really power hungry you should have no problem triggering two amps off the 1050. I would think you could split it at least 5 times and not bog down the 1050 drivers.

I don't know that leaving today's solid state electronics on versus power cycling them makes much difference. I'd give a different answer if it was a tube amp.