I know you can press the AT disconnect button on the sides of the throttle.
You can press the A/THR button on the FCU.
When does it disconnect automatically? For example, on landing.
The autothrust disconnect is described in the Flight Crew Operating Manual:
A/THR DISCONNECT
When the A/THR is disconnected, it is neither armed nor active. The A/THR can be disconnected in two ways:
- Standard disconnection
- The pilot pushes the instinctive disconnect pushbutton on the thrust levers (which immediately sets the thrust corresponding to the lever positions) or
- The pilot sets both thrust lever to IDLE detent.
- Non-standard disconnection
- The pilot pushes the A/THR pushbutton on the FCU while A/THR is armed/active, or
- The system loses one of the arming conditions.
(Airbus A320 FCOM - Autoflight - Flight Guidance)
On landing, the autothrust is actually not automatically disconnected. The Airbus will remind the pilots to retard the thrust levers to idle with the "RETARD, RETARD" callout when passing 20 ft (see e.g. this answer). Moving the thrust levers to IDLE will then disconnect the autothrust system.