My textbook (New Tertiary Mathematics, Volume 1 Part 1, Pure Mathematics: The Core, by C Plumpton & P S W Macilwaine) introduces Parameters in the following manner:
"The coordinates of a point on a curve can usually be expressed in terms of a third variable which is called a parameter. For example $(t^{3}, t^{2})$, where $t$ varies, are the coordinates of a point on the curve $y^{3} = x^{2}$."
I really don't understand this at all and I thought here might be the place to ask for help understanding it.
Perhaps I'm thinking about this wrong, but what I take from a sentence like that is: if I choose some value of $t$ - let's say $t = 2$ - I should get the coordinates of a point on the curve $y^{3} = x^{2}$. But if we try that with $t = 2$, we get $(8, 4)$. But if I try letting $x = 4$, I get $y = \sqrt[3]{16} \neq 8$.
So, if I may, I'd like to ask what I'm supposed to understand by that sentence.
The parametrization given is $(x,y)=(t^{3}, t^{2})$.
You have inverted the roles of $x$ and $y$, probably mislead by $y$ appearing first in the equation $y^{3} = x^{2}$.