Is it safe to put the lights in parallel to the power cable of the MP edge?
That's basically what the lighting circuit on the throttle switch does, as the battery feed wire going to the throttle switch is connected to the main battery wire inside the motor harness before it goes into the controller.
It might be possible to fit a simple on/off switch into your light unit if there is room, but on my MPII, I have my front light on all the time when the battery is turned on, so it acts like a daylight running light to make me more visible to other road users.
It is only a 4W LED lamp
(despite being advertised as 12W), so the current being drawn from my 29.6V battery pack is relatively low
(0.135A) which is ~2.23% of my 5.8Ah battery capacity, so it doesn't affect my typical 4 mile "dog walking" rides.
Alan