Neurological and psychiatric complications continue to be a public health concern in long coronavirus disease 2019 (COVID-19). This varies from olfactory dysfunctions such as parosmia to cognitive and emotional challenges. Historically, the surge of neurological disorders followed the viral pandemics, for example, the emergence of Encephalitis Lethargica after the outbreak of Spanish Influenza. During and after COVID-19 infection, the problems associated with the sense of smell and the reports of affected olfactory and limbic brain areas are leading to a growing concern about the similarity with the symptoms and the pattern of degeneration observed at the onset of Parkinson's disease and Alzheimer's disease. These reports reveal the essentiality of long-term studies of olfactory and cognitive functions in the post-COVID era and the experiments using animal models to dissect the neural basis of these complications. In this manuscript, we summarize the research reporting the potential correlation between neurological disorders and viral pandemic outbreaks with a historical perspective. Further, we discuss the studies providing evidence of neurodegeneration due to severe acute respiratory syndrome coronavirus 2 infection by focusing on viral Parkinsonism.