Is it possible to program ESP8266 from another microcontroller?
I assume you are compiling and uploading (running esptool) on the PC. What I would try is:

Connect the serial lines from the PL2303 to the ESP8266 via two series resistors (start with trying 1k). Keep the AVR (which I assume is also connected to the serial pins of the ESP) in reset while downloading to the ESP. When the AVR comes out of reset it can 'take over' the seria lines because its output is much stronger than the 1k-impedance signal from the PL2303.

Notes
- the AVR is connected directly to the ESP pins
- you only need a resistor in the Rx line of the ESP, but why no put one in both
- you might have to lower the esptool baudrate 
- you can do this in a more fancy way with a real switch, a digital switch, an analog switch, or even a relay