During the coronavirus disease 2019 (COVID-19) crisis, different personality characteristics may have influenced parental well-being in different ways. In the present study, we combined variable and person-oriented approaches and examined relationships between resilience, parental burnout, and perfectionism during the lockdown. We first used structural equation modeling to assess the paths between variables. We then used latent profile analysis to examine different profiles of parents based on resilience, perfectionism, and symptoms of parental burnout. Finally, we examined how these profiles differ in terms of relevant background variables. The results showed that resilience predicted parental burnout negatively even after controlling for multidimensional perfectionism. Parents’ age, children's age, children's special needs, and the increase in time spent with children due to lockdown contributed independently to burning out as a parent. Three profiles were found: a resilient profile, perfectionist profile, and burned-out profile. Resilient parents were likely to be men, older, and with less financial difficulties than parents in the other two profiles, and less likely to spend increased time with their children due to lockdown than the burned-out parents. Perfectionist parents, in turn, had older children than the burned-out parents did. These results suggest that resilience may help parents overcome burnout at times of crisis.