What can cause the gear shift to stick in the park position. on 2005 Chrysler 300

There was no indication that this would occur. The steering wheel will not lock when the ignition is off and the key is out of the ignition.

Asked by for the 2005 Chrysler 300
The Brake Transmission Shifter/Ignition Interlock (BTSI) is a cable operated system that prevents the transmission gear shifter from being moved out of PARK without the proper driver inputs. The system also contains a solenoid that is integral to the shifter assembly. The solenoid works in conjunction with the park lock cable to permit shifter movement out of PARK when the brake is depressed. The Brake Transmission Shifter/Ignition Interlock (BTSI) is engaged whenever the ignition switch is in the LOCK position. An additional electrically activated feature will prevent shifting out of the PARK position unless the brake pedal is depressed at least one-half inch. A solenoid in the shifter assembly is energized when the brake pedal is depressed. When the key is in the ON position and the brake pedal is depressed, the shifter is unlocked and will move into any position. The interlock system also prevents the ignition switch from being turned to the LOCK position, unless the shifter is in the gated PARK position. Here is a link to your local repair facility, however I recommend a visit to your local dealership for this repair.