Why if I'm parked and playing the radio, when I go to start the car the head unit restarts? Is there a way to fix this?

That's normal. It's because going from ACC to START cuts the power to the red switched ACC wire. It's kind of like a computer, you cut the power, it reboots.

The only way I can think of to fix this is maybe to get some kind of "off after delay" type of relay and wire it into the ACC wire.

a very easy way to fix this is just to find another power source. there are many things in your car that do not turn off when you try to start it. just find one and tap your wire to that. some things are cigarette lighters, domelights, headlights, horn, flasher lights, etc.

All of those things mentioned would be hot at all times.

Don't forget that in many cars, if you want to use the wire that's also hot in the ACC position, the ACC wire is really the only one you can use. I'm sure it depends on the car so getting ahold of a good wiring diagram would be helpful.

There are some circuits in the car that will be hot in ON/RUN/START without turning off while the car is being cranked; ECM, PCM, Ignition, and injector switched power feeds for example, but none of those are hot in the ACC position.

I wouldn't advise bypassing this safety feature. If your batteries run low and you try to start the car, the voltage on the circuit goes to squat and the amps freak. Most electronics don't do well in this environment. That's why lights, lighters and starters typically only function in start mode. If you really want to get around the issue properly, install an accessories battery that is isolated from the starting battery. That way you can run your radio any time you would like, for days and days, and not affect your starting circuit. Plus if you do try to start your car with a low or dead starting battery, you won't convert the AVIC to a blue flame generator.