Why are coordinates marked at parking bays at civil airports?
These checkpoint coordinates are used to verify the current location of the aircraft for navigation purposes. The coordinates can be used to check, input or calibrate the navigation system's current location. It can be used to input or calibrate the aircraft's Inertial Reference System (IRS) during initial alignment by providing a precise Latitude/Longitude location the pilot can input into the system when the aircraft is powered up.
The IRS uses 3 laser gyros to sense the forces created when the aircraft moves from this starting position, updating the IRS's position accordingly.
As you'd expect, due to drift, the IRS position will not remain accurate after crossing the Pacific Ocean. Once within range of ground-based navigation facilities, the aircraft's location is updated by the Flight Management System ("FMS" - the airplane's navigation computer) utilizing data from these facilities, such as VORs (many of which provide distance information via DME) which the computer uses to triangulate a position.
Information from ground-based navigation facilities is not available until airborne, so you need the Lat/Long gate location for INS initial alignment at the gate (especially if you were flying in 1990 or are currently flying an aircraft from the 1980s or early 1990s that is not yet updated with GPS capabilities).
The Flight Management System (FMS) navigation database of a modern airplane will typically contain the Lat/Long coordinates for various gate locations at major airports.
GPS (Global Positioning System) satellite position information can also be used by the FMS to update the aircraft's position.
On modern aircraft GPS provides an accurate Latitude Longitude position to your navigation system no matter where you are (including sitting at the gate). While these Lat/Longs at the gate are likely a bit of a holdover from the 1980s/1990s when you needed a Lat/Long position for your INS and could not get it from GPS, these coordinates at the gate are still valuable, giving you the ability to check the accuracy of your indicated GPS location or verify the gate position shown in the FMS navigation database (in addition to giving your IRS a current location for initial alignment).
These gate Lat/Long coordinates are also indicated on some Jeppesen airport diagram charts which show zoomed in gate details.
I just did a search to see if someone already has explained the IRS system here, and found an extremely thorough and detailed description. Check it out: Why are there two IRS switches on the 737NG?