Let $c(\pi)$ stand for the number of cycles in the unique disjoint cycle representation of $\pi.$
Consider $\pi \in S_n$ and $\pi = \tau_1 \tau_2\ldots \tau_a.$ Then $a\equiv n - c(\pi)\pmod 2$. In other words $a$ and $n - c(\pi)$ have the same parity "and so two different decompositions of $\pi$ into transpositions will both have an even or both have an odd number of terms."
I am a bit confused about how they derived the conclusion in the quote above. Can someone, please, elaborate a little.
For $\pi\in S_n$ let $f(\pi)=n-c(\pi)$; this really is a number that depends only on $\pi$ itself. Now suppose that we decompose $\pi$ as a product of transpositions in two ways,
$$\pi=\tau_1\tau_2\ldots\tau_a=\tau_1'\tau_2'\ldots\tau_b'\;.$$
The result says that $a\equiv f(\pi)\pmod2$ and $b\equiv f(\pi)\pmod2$, so $a\equiv f(\pi)\equiv b\pmod2$, and since congruence mod $2$ is an equivalence relation, this says that $a\equiv b\pmod2$. But that just means that $a$ and $b$ have the same remainder on division by $2$, so either $a$ and $b$ are both even, or $a$ and $b$ are both odd.