Yes, we even know that the density of such $n$ is the expected one.

A. O. Gelfond proved in 1968 ("Sur les nombres qui ont des propriétés additives et multiplicatives données", Acta Arithmetica 13, pages 259--265) that $$\lim_{N \to \infty}\frac{1}{N/a}\#\{ 1 \le n \le N: n \equiv b \bmod a, \, s_q(n) \equiv j \bmod m\}=\frac{1}{m}.$$ Here $a,q,m$ are fixed positive integers, $s_q$ is the sum of digits in base-$q$, and $\gcd(m,q-1)=1$ (otherwise there are obvious obstructions, due to the congruence $n \equiv s_q(n) \bmod {q-1}$).

Now apply this with $q=m=2$ to get your answer.

Yes, we even know that the density of such $n$ is the expected one.

A. O. Gelfond proved in 1968, in a short paper ("Sur les nombres qui ont des propriétés additives et multiplicatives données", Acta Arithmetica 13, pages 259-265) that, for all $b \bmod a$ and $j \bmod m$, $$(\star) \, \lim_{N \to \infty}\frac{1}{N/a}\#\{ 1 \le n \le N: n \equiv b \bmod a, \, s_q(n) \equiv j \bmod m\}=\frac{1}{m}.$$ Here $a,q,m$ are fixed positive integers, $s_q$ is the sum of digits in base-$q$, and $\gcd(m,q-1)=1$ (otherwise there are obvious obstructions, due to the congruence $n \equiv s_q(n) \bmod {q-1}$).

Now apply this with $q=m=2$ to get your answer.

Regarding the error term in $(\star)$: already Gelfond proved a power saving result. This means that the expression in the limit in $(\star)$ is $1/m$ plus $O_{a,m,q}(N^{-c})$ for some concrete $c$ depending on $m$ and $q$.

An optimal value for $c$ when $q=m=2$ was determined in 2009 by V. Shevelev ("Exact exponent in the remainder term of Gelfond's digit theorem in the binary case", Acta Arithmetica 136, pages 91-100).