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Background: Although smell and taste disorders are highly prevalent symptoms of COVID-19 infection, the predictive factors leading to long-lasting chemosensory dysfunction are still poorly understood. Methods: 102 out of 421 (24.2%) mildly symptomatic COVID-19 patients completed a second questionnaire about the evolution of their symptoms one year after the infection using visual analog scales (VAS). A subgroup of 69 patients also underwent psychophysical evaluation of olfactory function through UPSIT. Results: The prevalence of chemosensory dysfunction decreased from 82.4% to 45.1% after 12 months, with 46.1% of patients reporting a complete recovery. Patients older than 40 years (OR = 0.20; 95% CI: [0.07, 0.56]) and with a duration of loss of smell longer than four weeks saw a lower odds ratio for recovery (OR = 0.27; 95% CI: [0.10, 0.76]). In addition, 28 patients (35.9%) reported suffering from parosmia, which was associated with moderate to severe taste dysfunction at the baseline (OR = 7.80; 95% CI: [1.70, 35.8]). Among the 69 subjects who underwent the UPSIT, 57 (82.6%) presented some degree of smell dysfunction, showing a moderate correlation with self-reported VAS (r = −0.36, p = 0.0027). Conclusion: A clinically relevant number of subjects reported persistent chemosensory dysfunction and parosmia one year after COVID-19 infection, with a moderate correlation with psychophysical olfactory tests.