Why do airliners have to park so accurately?
This is to ensure that there is space for all equipment around the aircraft. It's not critical to the centimeter, but if it's off by a meter or more there may be problems. For example, the tail may stick out into the taxiway behind or the nose may stick into the service road in front of the aircraft. While most aircraft stop with the nose wheel on or very near the stop-mark, the main wheels can be off by a few feet, depending on how early/late the pilot turns into the stand. 

Modern aerobridges have a wide range of movement to cater to various aircraft types, but nowadays there is a lot of equipment buried in the ground. You might see ground power and air conditioning hoses which plug into the ground next to the aircraft nosewheel area. If the aircraft is off by a few feet the hose/wires may cause obstruction to the ground crew. The hydrants for fuel also need to be near the fuelling ports - too far and the hoses might not reach.