Multidimensional models of perfectionism postulate the existence of various perfectionism traits, with different effects on mental health. In order to suggest parsimonious targets in psychological interventions for university students, this study aimed to explore whether, how, and which individual perfectionism traits are uniquely associated with stress and well-being. The participants were 253 students aged 18–30 who completed the Frost Multidimensional Perfectionism Scale, the Perceived Stress Scale, and the Warwick–Edinburgh Mental Well-being Scale. Controlling for the common variance of perfectionism traits in statistical analysis, it was shown that (1) Personal Standards were associated with higher well-being and lower stress, (2) Concern over Mistakes and Doubts about Actions were related to lower well-being and higher stress, (3) Parental Expectations and Parental Criticism were not correlated with stress, and (4) Parental Criticism was associated with lower well-being. In the multi-predictor mediation model, with five perfectionism traits as predictors, perceived stress was a significant mediator between several perfectionism traits (i.e., Personal Standards, Concern over Mistakes, and Doubts about Actions) and well-being. Overall, Personal Standards, Concern over Mistakes, and Doubts about Actions seem to be parsimonious psychological targets, with Personal Standards expressing mental health-promoting effects, whereas Parental Expectations and Parental Criticism seem to be less important psychological targets.