Is this language in $PSPACE^{L'}$?
Yes it is, by exactly the argument you give. Looping over all possible strings of length&nbsp;$n$ requires only $n$ tape cells&nbsp;&ndash; you can do it by treating the length-$n$ strings as numbers in base&nbsp;$|\Sigma|$, so looping through those is essentially a counter, too. You can reuse the space since you don't care about strings that you've already tested. Testing if the string you just made is in&nbsp;$L'$ takes space&nbsp;$n$ on your oracle tape. Implementing the counter requires only $O(n)$&nbsp;bits, since you only need to store a number in the range $0, \dots, |\Sigma|^n$. So the total amount of space required is linear in&nbsp;$n$.

In fact, you can do slightly better, though the total will still be $O(n)$ because of the need to loop through the length-$n$ strings.  You don't need to count how many strings in&nbsp;$L'$ you've seen so far; you just need to know if that number is odd or even.