First we may notice that
$$\sum_{i=0}^m \binom{n}{i} = [x^m]\ \frac{(1+x)^n}{1-x} \equiv_2 [x^m]\ \frac{(1+x)^n}{1+x} = [x^m]\ (1+x)^{n-1} = \binom{n-1}m,$$
which was already pointed out by @FedorPetrov in the comments.

Then modulo 2 the given expression is congruent to
$$\binom{j-1}{2j-k-1} + \sum_b \binom{b-1}{2b-k-1} \binom{j-b-1}{2j-(2k+1)+b}$$

Next, we notice that $\binom{b-1}{2b-k-1}=\binom{b-1}{k-b}$ equals the coefficient of $x^{k-b}$ in $\frac{C(-x)^{-k}-(-xC(-x))^k}{\sqrt{1+4x}}\equiv_2 C(x)^{-k}+(xC(x))^k$, where $C(x):=\frac{1-\sqrt{1-4x}}{2x}$ is the generating function for Catalan numbers.

Similarly, $\binom{j-b-1}{2j-(2k+1)+b}$ is congruent to the coefficient of $x^{2j-(2k+1)+b}$ in $C(x)^{-(3j-2k-1)}+(xC(x))^{3j-2k-1}$. 
Noticing that $3j-2k-1<k$, we conclude that the sum in the last expression is congruent to the coefficient of $x^{2j-k-1}$ in 
$$C(x)^{-(3j-k-1)}+(xC(-x))^{3j-k-1}+x^{3j-2k-1}C(x)^{3j-3k-1}+x^{k}C(x)^{3k-3j+1},$$
that is $\binom{j-1}{2j-k-1}+\binom{2(k-j)}{k-j}$. Hence,
$$\binom{j-1}{2j-k-1} + \sum_b \binom{b-1}{2b-k-1} \binom{j-b-1}{2j-(2k+1)+b}\equiv_2 \binom{2(k-j)}{k-j},$$
which is congruent to 1 iff $k=j$.