If the cough has persisted for a couple weeks, the it is more likely to be allergies. It is true that there can be some persistent inflammation of the airways after resolution of a cold, so that there is some persistent cough that will commonly last for several days, but can last for a week or two. Since you are at the outer limit of this time and it is apparently still persisting, then it is more likely to be allergies.
I actually suffer from the same problem, in that I have winter allergies, which are not too bad unless and until I get my first cold of the season, after which my allergies will flare-up significantly and continue to be symptomatic for several weeks or a month or two. It is never so bad that I need to treat it very aggressively, but it can be irritating for a couple months.
Like other allergy symptoms, it would be appropriate to first try an antihistamine
for the cough, but if the symptom persists, it would be appropriate to also use a nasal spray. Doctors would prefer to use a steroid nasal spray, but these are only available by prescription. On the other hand, NasalCrom, which is available over the counter, would be an easier option. Cough suppressants are also OK, but they tend to not work as well for a cough from allergies as for a cough from a cold.
If you have any further questions, please let me know.