Local existence of parallel vector field

@John Ma's answer is a good one, but I'd also like to point out that parallel vector fields are even rarer that Killing vector fields. For example, on a round sphere, there are plenty of Killing vector fields but no nontrivial parallel fields. Basically, the existence of a parallel vector field is equivalent to the condition that the metric splits locally into a Riemannian product of a one-dimensional manifold and an $(n-1)$-dimensional one. This implies, in particular, that the sectional curvatures of planes containing $V$ are all zero.

This is too strong a condition. $\nabla V = 0$ would imply that $V$ is a Killing vector field, thus the local one parameter subgroup of diffeomorphisms are isometries. As isometry preserves curvature, one can construct examples so that such a local vector field can't be found. For example, one can take a surface so that it's Gauss curvature attains a strict maximum/minimum at $p$.