Spinor fields are sections of a spinor bundle, so you have to be careful when you work with them as if they were functions. For a spinor field, the notions of being parallel, Killing, conformal Killing,... make perfect sense globally as equations on sections of the spinor bundle, but you have to specify which bundle.

Spinor bundles are associated vector bundles to a spin bundle, which is a lift of the oriented orthonormal frame bundle. Lifts need neither exist or, if they do exist, be unique, hence there are manifolds on which you cannot define a spin bundle and manifolds on which you have more than one such bundle. The obstruction for the existence of a spin structure is orientability and the vanishing of the second Stiefel-Whitney class of the tangent bundle. If a manifold M admits a spin structure, then it may admit more than one: they are classified by $H^1(M,\mathbb{Z}_2)$ which is isomorphic to the set of group homomorphisms from the fundamental group to $\mathbb{Z}_2$. Roughly speaking this measures how you can consistently assign signs to noncontractible loops.

The circle has fundamental group $\mathbb{Z}$ and since there are two homomorphisms $\mathbb{Z} \to \mathbb{Z}_2$, there are two different spin structures, which in string theory are usually called NS and R, much to the amusement of spin geometers everywhere.

Hence the lesson is that before you can even talk about `<insert your favourite spinor equation>`

you need to say what your spinors are; that is, which spinor bundle they are sections of.

A rough analogy (which can be made precise in this case) is that you have equations and then boundary conditions and both are necessary in order to define the problem. The analogue of the boundary conditions is specifying the spinor bundle. This is indeed the case for the circle: where the spinor field will either change by a sign or not as you move along the circle.

It is not uncommon for manifolds admitting inequivalent spin structures, that there should be parallel, Killing,... spinor fields relative to one of the spin structures, but not relative to others. In fact, this is the generic situation.

In summary, the answer to the question in the title is emphatically **Yes**.

**Further remarks**

This may answer the OP's question in the comment to an earlier version of this answer.

One has to be careful to conclude that a spinor field does not obey the right periodicity conditions. Indeed, one must remember that there is a "gauge" symmetry whenever one deals with sections of associated vector bundles to principal bundles, and that is the freedom to perform a local $G$-transformation, where $G$ is structure group of the bundle. In the case of the spinor bundles, the structure group is the relevant Spin group. Hence it could be that the discrepancy in sign is simply an artefact of the choice of frame and can be cured by a local Spin transformation. With apologies for referring to my own work, an illustrative example of this occurs at the end of §3.2.2, particularly around equation (32), in my paper with Gutowski and Sabra on 4- and 5-dimensional preons: arXiv:0705.2778 [hep-th].

I hope that this helps.

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