Parallel sections are invariant under parallel transport along all holomorphic disks, so if holomorphic somewhere, they are holomorphic everywhere. The kernel of the connection is a linear subspace of the holomorphic sections, so a linear system. If you start with a nonzero holomorphic section of a holomorphic vector bundle, you can construct local holomorphic connections which preserve it, in some neighborhood of any chosen point.
Edit: each parallel section is uniquely determined by its initial value at a point. As you travel around loops, it undergoes some monodromy. But the set of initial values at a point for which there is a parallel section is a vector subspace $F_{x_0}$ of the fiber $E_{x_0}$ at that point $x_0$. That fiber $F_{x_0}$ goes around in parallel inside $E$, making a vector bundle $F$ with flat connection.
Edit: When the connection is real analytic, we can parallel transport along any path, analytically continuing any given local parallel section, as linear ODE always have solutions. Analyticity ensures that the resulting local extension of a parallel section remains parallel. So all parallel sections extend to the universal covering space. Hence the fibers $F_{x_0}$ are indeed all of the same dimension, and carried invariantly under parallel transport. For $C^{\infty}$ connections, this won't work; you can have local parallel sections which do not extend over "bumps", and $F_{x_0}$ will change dimension.