Simple odd/even question

If you restrict the domain to $0 \le t \le 2\pi$, then $y=t^2$ is neither odd nor even as it's not even defined for -ve $t$.

However, you can create a periodic function on $\mathbb{R}$, by repeating the graph you have on the restricted domain, either via an odd extension or an even extension. In the picture you gave, the function has an neither an even nor odd extension since neither $f(-t)=f(t)$ nor $f(-t)=-f(t)$ are true.

I guess this is something to do with Fourier series?

Yes this is from a lecture on Fourier series. One thing I am confused about: if I was told to sketch f(t) = t^2, and 0<t≤2pi and I sketched the following (without repetition) would I still yield the correct answer if I deduced whether it is even or odd based on this non-repeated form?