The definitions of even and odd that I'm familiar with have to do with expressing a permutation as a product of transpositions.
Specifically, every permutation $\sigma$ can be expressed as a product of transpositions $\tau_1 \tau_2 \cdots \tau_n$. Then $\sigma$ is even if $n$ is even, and $\sigma$ is odd if $n$ is odd, where $n$ is the number of transpositions appearing in the product.
The problem is that there are lots of ways of expressing a permutation as a product of transpositions. For example
$$(1~~2~~3) = (1~~3)(1~~2) = (2~~1)(2~~3) = (1~~2)(1~~3)(2~~3)(2~~1)$$
So you need to show that an even permutation $\sigma$ doesn't magically become odd just by expressing it in a different way as a product of transpositions—that is, you need to show that if some expression of $\sigma$ as a product of transpositions has an even (resp. odd) number of transpositions in the product, then all such expressions do.