Pull signal low using a pic microcontroller
The ADH8066 may have an inbuilt resistor that is a pull-up to it's supply voltage. You'll need to understand if this is so and if not you'll need to apply one. Then use an NPN BJT with emitter to 0V / ground. Connect the collector to the on/off pin on the ADH8066 and connect the base (via a 10k resistor) to the PIC pin you want tu use for controlling on/off.

A high coming from the PIC pin will cause the NPN to switch on and pull the relevant GSM module's pin to ground/0V. A low on the PIC pin will turn off the NPN and the external pull-up (or internal) will pull the module's control pin to Vcc.