A320 Captain reported push crew failure resulted in aircraft sharklet making contact with another A320 elevator resulted in damage and tow back to gate.

Narrative

Aircraft X Event occurred while operating on Flight XXX (ZZZ-ZZZ1) upon pushback from Gate XX at ZZZ Airport. Upon completion of the before start checklist; we initially received clearance from Ramp Control to the middle of the alley; which was then revised by Ramp Control to continue to the top of the alley to Spot XX. While under tow; and while being towed backwards down the alley; our left Sharklet contacted the right elevator of Aircraft Y parked at Gate XY. Aircraft Y advised Ramp Control that their aircraft shuddered with possible contact with our aircraft. We immediately called the tug driver to stop the aircraft to analyze the situation. Upon Aircraft Y Captain's inspection; it was noted that Aircraft Y was contacted with visible damage to the outer most portion of the right elevator. I then contacted the Chief Pilot for further guidance. We then decided; with Chief Pilot's concurrence; to return to the gate to have Maintenance perform a more detailed inspection of the damage. Upon inspection of our aircraft; it was discovered that the fourth static wick from the top of the sharklet was missing with surface damage to the rear edge of the sharklet and minor scratches to the outer surface. Once blocked in; I contacted both the Dispatcher and Operations to notify them of the gate return. Our aircraft was configured for push; with APU operating and both engines off. Event occurred approximately at XA45 local; XA45 UTC; on Date.