B757 First Officer reported while parking the aircraft and shutting down the engines the aircraft continued to move forward resulting in the air crew losing control of the aircraft and hitting ground support equipment.

Narrative

During parking of aircraft at Gate XX in ZZZ; brakes were very touchy in the fact that the aircraft seemed like it was on hydraulics with a slight bounce. Once at a complete stop; Captain pulled the parking brake and pointed to the EICAS to verify messages with finger and overhead panel to complete flows to shut engines down. The First Officer verified the parking brake was pulled; but could not see the parking brake light from their seat due to the thrust control panel blocking vision. Right engine was shut down while left engine was still running and being completed for shut down when Captain realized first that the aircraft was still moving and slammed on the brakes. After the First Officer noticed the Captain pulling the parking brake handle and confirming by pointing to the EICAS; the First Officer (FO) was looking down and away at the tough pad iPad to shut it down. FO also saw aircraft moving out the corner of their eye when the Captain was already executing immediate stoppage of the aircraft via the brakes. Unfortunately; stopping measures by the crew were not fast enough and the aircraft collided with the GPU on the right engine cowling and was told by Maintenance that the nose wheel ran over the ramps only tow bar and possibly damage to the nose wheel. All ground personnel were ok and safe after the incident occurred.Cause: Unknown whether the brakes failed or the parking brake lever released.Suggestions: First Officer could have been more focused outside the aircraft to make sure the aircraft isn't moving and waiting for the Captain to finish his shutdown checklist before completing any items on the First Officers shutdown checklist. Also could have double and triple checked to make sure the parking brake light was illuminated by the parking brake handle even though the light is blocked by the thrust panel and making sure the EICAS kept a parking brake message for more than a second.