These are more general properties of the parity which I think, they give a better explanation of what this symmetry is really about.
The general definition of parity is an operator $\mathcal{P}$ with the properties $\mathcal{P} = \mathcal{P}^*$ and $\mathcal{P}^n=\mathbb{1}$, $*$ denotes complex conjugation. Most of the time, people stick with $\mathcal{P}^2=\mathbb{1}$. Perhaps it is also easier to look parity in a discrete Hilbert space in a one-dimensonal system. For this it is enough to look for mirror symmetry. That is, consider a tight-binding chain in which each site is described by the basis
\begin{equation}
\lbrace |\psi_1\rangle,\dots,|\psi_N\rangle\rbrace.
\end{equation}
The parity $\mathcal{P}$ shoudl act on this basis as $|\psi_1\rangle\rightarrow|\psi_N\rangle$, $|\psi_2\rangle\rightarrow|\psi_{N-1}\rangle$, and so forth. Fixing the basis, you can write the parity operator as
\begin{equation}
\mathcal{P} = \begin{pmatrix}
& & & 1 \\
& &1& \\
&\unicode{x22f0}& &\\
1 & & &
\end{pmatrix},
\end{equation}
all the other entries are zero. Furthermore, this representation is also suitable to represent parity in discrete models which are not one-dimensional. For example benzene, ethylene etc.
Finally, from my point of view, parity is explicitly put into action when you integrate relativity in quantum mechanics, i.e. in Quantum Field Theory. There, the position is $x^\mu = (t,\vec{x})$ and fixing a metric, the parity operator is given by the matrix
\begin{equation}
\mathcal{P}^\mu_{\;\nu} = \begin{pmatrix}
1 & & & \\
&-1 & &\\
& & -1 & \\
& & & -1
\end{pmatrix}.
\end{equation}
Then $\mathcal{P}: (t,\vec{x}) \rightarrow (t,-\vec{x})$, which amounts to a matrix multiplication $\mathcal{P}x$.
As a bonus, returning to non-relativistic quantum mechanics, people have studied a lot parity-symmetry Hamiltonians for the past 10 years or so.
