Hi,
Spanish cities have their own local police and local traffic law (of course, based in a national system). The point is that I don't think Gijon city hall can manage to send the fine to your house, so probably you won't pay it.
I'm not absolutely sure about that because I know there is an European agreement to share information about drivers who make offenses in the European territory, but I think city halls are out of this systems.
No body will chase you. In Gijon, as any Spanish city, operates a local police which is in charge of the city traffic and, out of cities, the Guardia Civil operates as a national police. (there are some exceptions as very small cities).
In a usual way they will send to your home the fine and give you a specific period of time to claim (is apply), and usually you should pay before this period expire (providing you don't file a claim). If you pay soon (don't know how soon), you could get a discount. Usually you pay "soon" by using the parking machines to at the city hall facilities.
I would recommend, if you don't want the discount, just wait to receive the fine at your home and then proceed.