Despite the importance and many applications that parametric equations and curves have in mathematics and other sciences, there is a lack of research regarding this topic in mathematics education. The goal of this article is to explore university students' understanding of parametric equations and curves based on two lenses: Action, Process, Object, Schema (APOS) and Onto-Semiotic Approach (OSA). Recent research showed that these two theories complement each other and the combination of them has good potential to analyse students' performance of mathematical concepts. Three tasks of parametric equations and curves were proposed to characterize the schema development of twenty-seven university students in terms of triad (i.e. intra, inter and trans). Based on the results, most of the students participating in this study had difficulties regarding the practical tasks and developing the mental structures needed to solve parametric equation tasks, especially the mental structures that have to be made to draw parametric curves and identify their directions, and to compute the first and the second derivatives of parametric equations. Finally, we finish the article by making practical suggestions for instructors and researchers.