Background: Fear of COVID-19 leads to stress and may result in various kinds of mental health problems. Many factors are associated with an individual’s perception of stress, including neuroticism and perceived social support. This study aimed to examine the role of neuroticism and perceived social support as mediators of fear of COVID-19 on perceived stress. Methods: Data from 3299 participants aged ≥18 years from the HOME-COVID-19 survey in 2020 were used for analysis. Measurements used included the Fear of COVID-19 and Impact on Quality of Life Scale, the Perceived Stress Scale-10, the Neuroticism inventory and the Multidimensional Scale of Perceived Social Support-12. A parallel mediation model within a structural equation modeling framework with 5000 bootstrapping sampling was used to test the mediating effect. Results: Fear of COVID-19 had a direct effect on perceived stress (B = 0.100, 95% CI = 0.080–0.121, p < 0.001), whereas neuroticism, but not perceived social support, partially mediated the relationship between fear of COVID-19 and perceived stress (B = 0.018, 95% CI = 0.000–0.036). Among all types of social support, only perceived support from friends was a significant mediator (B = 0.016, 95% CI = 0.006–0.025). Conclusions: Neuroticism and perceived support from friends are critical factors in the relationship between fear of COVID-19 and perceived stress.