Can I use serial communication for audio amplifier remote turn on?
Without more information, it's hard to say exactly, but you'll have to have some more design behind this to make it work as you want. The Tx and Rx lines of a serial port will be constantly moving between HIGH/LOW. If you have a scope, you could have a look at what kind of activity comes out on them.
The serial line is not designed to provide power, and I wouldn't recommend it. It would make sense to have a small timer circuit that looks for a HIGH/LOW transition on the serial line at some frequency (say, at least every 10 seconds) and produces a signal which can turn on a transistor to feed your amplifier. If the device is off, and it hasn't produced such a transition, the signal would be powered off. A 555 timer chip could do this as a &quot;missing pulse detector&quot;.
The PAM8403 has a &quot;Shutdown Control&quot; input, which can just take a digital input to tell it to turn off. The datasheet provides the following information:

Shutdown Operation
In order to reduce power consumption while not in use, the PAM8403 contains shutdown circuitry to turn off the amplifier's bias circuitry. This
shutdown feature turns the amplifier off when logic low is applied to the SHDN pin. By switching the SHDN pin connected to GND, the PAM8403
supply current draw will be minimized in idle mode. The SHDN pin can be left floating due to the internal pull-up.

So you should be able to put your signal from the 555 to there to put the amplifier into a low power state.