Transpositions generate $S_n$

Cycles. Every permutation decomposes into disjoint cycles.

Cycle decomposition. A $k$-cycle $(a_1\,a_2\,\cdots\,a_k)$ equals
$$(a_1\,a_k)(a_1\,a_{k-1})\cdots(a_1\,a_2).$$
So each cycle is a product of $k-1$ transpositions.

Conclusion. Every permutation is a product of transpositions. Hence transpositions generate $S_n$.

Parity. A permutation is even iff it decomposes into an even number of transpositions; this parity is well defined.