This dissertation reports on an investigation of fifteen second-semester calculus students' understanding of the concept of parametric function, as a special relation from a subset of R to a subset of R2. A substantial amount of research has revealed that the concept of function, in general, is very difficult for students to understand. Furthermore, several studies have investigated students' understanding of various types of functions. However, very little is known about how students reason about parametric functions. This study aims to fill this gap in the literature. Employing Action--Process--Object--Schema (APOS) theory as the guiding theoretical perspective, this study proposes a preliminary genetic decomposition for how a student might construct the concept of parametric function. To determine whether the students in this study made the constructions called for by the preliminary genetic decomposition or other constructions not considered in the preliminary genetic decomposition, data is analyzed regarding students' reasoning about parametric functions. In particular, this study explores (1) students' personal definitions of parametric function; (2) students' reasoning about parametric functions given in the form p(t)=(f(t),g(t)); (3) students' reasoning about parametric functions on a variety of tasks, such as converting from parametric to standard form, sketching a plane curve defined parametrically, and constructing a parametric function to describe a real-world situation; and (4) students' reasoning about the invariant relationship between two quantities varying simultaneously when described in both a graph and a real-world problem. Then the genetic decomposition is revised based on the results of the data analysis. This study concludes with implications for teaching the concept of parametric function and suggestions for further research on this topic.