NC-Expy, where did you get power from? How big is your amp (W RMS total)? It sounds like to me that you are either getting the power from a spot that is being feed by a small gauge wire or the wire between the source of power and the amp is too small/has a high resistance connection. In your case, you are right on the edge and what is happening is when the truck is running, the alternator maintains the voltage at 13.7-14.4 VDC. But, turn off the truck and the voltage drops to 12.6. Most amps have a cut off that if the input voltage drops below about 10 VDC, the amp goes into protection mode to prevent pulling excessive current through the DC-DC converter.
You can confirm the low voltage protection being the source of your cutting in and out by simply playing the music at low volume, seeing what the music does and then playing the same music, but at a much higher volume. If it is cutting out at high volume and not at low volume, then you need to find your too small gauge wiring/high resistance connection.
if you need more help with this, let me know. I will need to know:
1) where you are getting your power from (be very detailed)
2) what gauge wiring you are using between the point you are picking the power off of to the amp
3) what size amp you have (W RMS total please in your current configuration, worst case, model of amp and whether you are running 4 ohm subs, 2 ohm subs, in parallel with each other, on separate channels and I will figure it out from there)
4) do you have an inline fuse installed and if so, what size is the fuse.
How are you with using a multimeter. If it is a high resistance connection, this is going to be a very vital piece of equipment unless you are willing to pay someone to fix it for you.