Dimension of the space of harmonic one-forms on compact manifold

Question

It is shown by Hodge theory that the space of k-harmonic forms are isomorphic to the space of k-cohomology class, and for harmonic one-forms on compact manifold we also know that they must be parallel ($\nabla X = 0$, where $X$ is its dual vector field). But if two harmonic one-forms coincide at a point p ($X_p = Y_p$), then at any other point q, let $c(t)$ be a path joining p to q (we also assume the manifold is connected here), we then have $\nabla_\dot{c} X = \nabla_\dot{c} Y = 0$, but since $X$ and $Y$ coincide at p, by unique parallel transport, they must also coincide at q. Since q is arbitrary, $X$ must coincide with $Y$ everywhere, but this shows that the dimension of the space of harmonic one-forms is less than or equal to the dimension of the manifold, which yields a contradiction when one considers a compact surface with many genuses, whose first cohomology classes could have dimension much greater than 2. I wonder what is the problem in my reasoning?