Parity conjecture for $\beta_{-N}$ when $a_E(2)$ is even

Let $E$ be an elliptic curve with conductor $N_E$, and let $a_E(2)$ denote its eigenvalue at $2$. For a prime $N\equiv 3\pmod{4}$, let $\beta_{-N}$ be the associated integer. Parity conjecture. If $N_E\equiv 3\pmod{4}$ and $a_E(2)$ is even, then $\beta_{-N}$ is even.

The numerical data in the table suggest this parity relation for the coefficients attached to elliptic curves of conductor congruent to $3$ modulo $4$; its general validity is left open.