Parallel LCD screen to serial? (1 dedicated microcontroller?)
Well, here's the pin out: -



And, the pins that need control are DB0 to DB7, RS, R/W and E.

You can use a serial in parallel out shift register (such as 74xx595) - this means you load the data word (DB0-7) using one pin for data and another pin acting as a clock for the serial register.

The other three pins are probably best left as dedicated IO pins so, in total you can get away with 5. That's a lot better than 11.