  We show that $3$-$(\alpha,\delta)$-Sasaki manifolds admit solutions of a
certain new spinorial field equation (the $\mathcal{H}$-Killing equation)
generalizing the well-known Killing spinors on $3$-Sasakian manifolds. These
$\mathcal{H}$-Killing spinors have more desirable geometric properties than the
spinors obtained by simply deforming a $3$-Sasakian metric; in particular we
obtain a one-to-one correspondence between $\mathcal{H}$-Killing spinors on
dual pairs of homogeneous $3$-$(\alpha,\delta)$-Sasaki spaces. Finally, we show
that $\mathcal{H}$-Killing spinors generalize certain special spinors in
dimension $7$ previously constructed by Agricola-Friedrich and Agricola-Dileo
using $\text{G}_2$-geometry.
