Background: Globally, the coronavirus disease-19 (COVID-19) pandemic has had a devastating psychological impact on people, especially the healthcare workers/students, in many different community settings. Limited research has been reported on the mental health issue of healthcare students in Vietnam. Objectives: This study investigated and quantified depression, anxiety, and psychological distress levels among healthcare students, with an emphasis on comparing those studying pharmacy to students in other healthcare-related disciplines, in Vietnam during the COVID-19 pandemic. Methods: A cross-sectional study recruited 2246 respondents between September 7 and October 7, 2021, in which 230 were pharmacy students with a mean age of 20.0 ± 1.6 years and most of them had a low COVID-19 fear level. The standardized Patient Health Questionnaire–4 (PHQ-4) scale, consisting of two subscales of PHQ-2 and Generalized Anxiety Disorder-2 (GAD-2), was utilized for the screening of depression and anxiety levels. Results: Pharmacy students scored significantly higher on the PHQ-4 psychological distress measure than non-pharmacy students (mean 2.23 vs. 1.90, p = 0.039). Compared to non-pharmacy students, pharmacy students had higher rates of anxiety (10.4% vs. 6.5%, p = 0.028). Conclusion: The prevalence of anxiety was higher among pharmacy students in comparison to non-pharmacy students. These findings present a need to recognize and conduct early intervention measures for mental health problems in healthcare students, especially pharmacy students, during recovery from the COVID-19 pandemic.