Path Parity

You are given an integer $N$. Let $A$ be an $N \times N$ grid such that $A_{i, j} = i + N\cdot(j-1)$ for $1 \leq i, j \leq N$. For example, if $N = 4$ the grid looks like:  You start at the top left corner of the grid, i.e, cell $(1, 1)$. You would like to reach the bottom-right corner, cell $(N, N)$. To do so, whenever