Which commercial aircraft can perform a parallel engine start?
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.