There is an interesting relation, as the \(n\)lab says. I'll try to explain it as informally as possible.

Let \(\mathcal{M}\) be a pseudo-Riemannian manifold. Then, a Killing vector field on \(\mathcal{M}\) is a *covariantly constant* **vector field** on \(\mathcal{M}\), and "pairing two covariant constant spinors (parallel spinors, i.e., Killing spinors with \(\lambda=0\)) to a vector yields a Killing vector". Similarly, a Killing tensor on \(\mathcal{M}\) is a *covariantly constant* **section of \(\mathrm{Sym}^k(\Gamma(\mathrm{T}(\mathcal{M})))\)**. Therefore, you may interpret ``Killing'' as being synonymous with ``covariantly constant'' (at least in these three cases).