Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.1 Parametric Equations - Exercises - Page 603: 18

The arrows indicate the direction of motion.

Since $x=t^2$, $x\geq 0$. So, we have $t=\sqrt{x}$. Substituting $t$ into $y$ gives $y=(\sqrt{x})^3=x^{3/2}$ Since $x$ is positive, $y$ is increasing. The direction of motion is going up.