$\binom{\ell}{a_1,\dots,a_k}$ is the coefficient of $x_1^{a_1}\cdots x_k^{a_k}$ in the expansion of $$(x_1 + x_2 + \dots + x_k)^{\ell}.$$ The sum of all these coefficients is obtained by substituting $x_1=\dots=x_k=1$.
To eliminate even $a_1$, we can consider the expansion of $$\frac{1}{2}(x_1 + x_2 + \dots + x_k)^{\ell} - \frac{1}{2}(-x_1 + x_2 + \dots + x_k)^{\ell}.$$
Continuing this way, we eventually get $$s(\ell,k) = \frac{1}{2^k} \sum_{t_1,\dots,t_k=0}^1 (-1)^{t_1+\dots+t_k} ((-1)^{t_1}+\cdots+(-1)^{t_k})^{\ell}$$ $$=\frac{1}{2^k} \sum_{z=0}^k \binom{k}{z} (-1)^z (k-2z)^{\ell}.$$
P.S. This formula resembles one for Stirling number of the second kind (formula (10) at MathWorld) but not quite.