Is it possible to connect two identical LCDs to display same information on both? One will be local and one on the remote box connected with cables. Would it work if I just solder them in parallel or there's some trick to it? 
It will work if the two LCDs are near the Arduino. The capacitance of the cables going to the remote LCD might cause a problem. I would start out with just the one (remote) LCD and get that working first, then add the local one.
Don
How long will the cable be? I personally only tested 18 inches with 40-wire ribbon cables to work properly. The cable is prone to interference such as turning on or off a halugen desk lamp nearby with put random characters on the display. I think the key is to use shielded cable. If that is too difficult for you with that many connections, use serial LCD. Only one data line to shield and you may drop baud rate in case the cable capacitance is high, i.e. cable too long.
You could use the sr library for the displays a 74HC595 is used to send serial data to the LCD's shielding 5 wires is a lot easier than 8, 4 data 2 control and 2 power. Longer unshielded cables can be used. On one I did several month's ago I had 12 inch cables with similar issues and I ran raw voltage to a local 5V reg and I used 470 0hm pulldowns At the LCD head end of the cable. The long wires to the displays pick op noise from "Stuff" in the room where they are being used... like liudr's halogen lamp.
Low value pull down resistors smooth out the noise by forcing the threshold where noise can be an issue to a much higher value as it takes more noise voltage to interfere with the lower impedance control lines.. Perhaps this will help, If it takes 1 volt to 'interfere' with a wire with a 4K7 pulldown, it would take 10V to 'interfere' with a 470R resistor pulldown. 470R will use 10 mA current (of a 40 mA max) from each port pin. 4K7 will use 1 mA from each port pin. With the exception of dedicated line driver ic's (which use similar methods) this is the easiest method. One of the great things is that it will only take 20 minutes of your time to test it. I would then try the Serial Shift register idea.
Doc