Perhaps you had the same misunderstanding that I initially did: I thought that the scenario was supposed to be a perfect clear with exclusively T pieces, but it actually allows any combination of other pieces. The answer is placing five T pieces in a way that clears the line containing their "odd color" tiles, then placing five L pieces to fill the remaining gaps.
Yes What happens is that blocks dropping from above a cleared line counteract the parity problem created by the odd number of T's. 5 blocks (an odd number) switch parity by moving one position orthogonally.
More generally, any odd number of line clears will reverse the parity of all remaining blocks above, which can be an odd number.
