Which commercial aircraft can perform a parallel engine start? In various Youtube videos, e.g. this one, one can see that the Boeing 787 is capable of a parallel engine start (both engines are started at the same time). Are there any other commercial aircraft around that allow for a dual (or multiple parallel) engine startup? 

What is the reason the Boeing 787 can perform parallel engine starts? What is the difference with other aircraft that cannot perform a parallel startup?
The main reason behind this capability is that the Boeing 787 starts its engines entirely on electrical power generated by the APU's VFSG (Variable Frequency Starter Generators) vs relying on APU bleed air to start the engines. Thus there isn't the typical drop in pneumatic pressure caused by the pneumatic starters on regular jetliners.
See 787 No-Bleed Systems for reference.

The Airbus A-340 can start both engines on a wing at the same time (1 &amp; 2 then 3 &amp; 4) as described here.