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.