What is this device they bring on board from the jet bridge when we park?
It is a temperature probe for the preconditioned air (PC air) unit. 

Bleed air is required in order to operate the air conditioning packs on an aircraft*. Thus, either an engine or the auxiliary power unit (APU) must remain running, burning jet fuel.  PC air pumps heated or cooled air into the aircraft and allows the pilot to shut down the engines and APU to conserve fuel.

This is what the system looks like from the outside:



A probe (located in the passenger loading bridge) is inserted into the cabin to regulate the air temperature while PC air is hooked up and running.

Upon parking, ground crews will typically hook up both ground electrical power and PC air, especially if the aircraft will be parked for any length of time. Since the air hose takes some time to unroll and roll, PC air may not be used if the flight is a "quick turn" and the APU will be left running.

* Except for the Boeing 787, which has a no-bleed system.