First, The Key FOB is not powerful enough to create interference like that.
Where did you ground your HIDs?
What Brand of Ballasts, Load Resistors, and Relay Harness did you use?
I'm assuming this is on your 05 Ram 1500 QC, if not specify year and model...
I think that you might be short circuiting between your parking lights and headlights, so you might want to check to see if any wires are corroded and/or damaged that are shorting to your HID relay.