Answer
FALSE
Work Step by Step
The first pair of equations only gives the right half of the parabola, that is, $y=x^{2}$. The second pair of equations traces out the whole parabola.
For example:
First case: when $x=t^{2}, y=t^{4}$
Then $(x,y)=(1,1)$ at $t=1$ and $(x,y)=(1,1)$ at $t=-1$
Second case: when $x=t^{3}, y=t^{6}$
Then $(x,y)=(1,1)$ at $t=1$ and $(x,y)=(-1,1)$ at $t=-1$
Thus, the second pair of equations traces outs the whole parabola, that is, $y=x^{2}$.
Hence, the given statement is false.
