= Solution
By <linearity> of the <time-independent Schrödinger equation>, the even and odd parts
$$
\psi_+(x)=\frac{\psi(x)+\psi(-x)}2,\qquad \psi_-(x)=\frac{\psi(x)-\psi(-x)}2
$$
each satisfy the same <energy eigenvalue> equation whenever they are nonzero. They obey $R\psi_+=\psi_+$ and $R\psi_-=-\psi_-$, so they have definite <parity>. At least one is nonzero, and $\psi=\psi_++\psi_-$. Each nonzero part can be normalized separately.
Therefore \b[a basis of stationary solutions can be chosen to have definite <parity>], and solving the even and odd sectors loses no <energy eigenvalues>. This does not assert that every possible <quantum superposition> has definite <parity>; a superposition of the two sectors need not. For one-dimensional <bound states> in the usual regular setting, <nondegeneracy of one-dimensional bound states> further implies that each individual <energy eigenstate> has definite <parity>.
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