Parity conjecture for $\beta_{-4N}$ and $\beta_{-4}$ when $N\equiv 1\pmod{4}$

Let $N$ be a prime congruent to $1$ modulo $4$, and let $\beta_{-4N}$ and $\beta_{-4}$ be the associated integers. Parity-matching conjecture. If

$$N\equiv 1\pmod{4},$$

then $\beta_{-4N}$ and $\beta_{-4}$ have the same parity.

The claim is suggested by the numerical data in the cited table. No proof or disproof is supplied in the text.