The covariant constant 1-forms are parallel. The value of a covariant constant 1-form is determined throughout each connected component by its value at any one point: just take that value and parallel transport along any path to any other point. This recipe will succeed in producing a 1-form just when the 1-form you start with is invariant under the holonomy group of the connection; otherwise there is no covariant constant 1-form with the given initial value. However, if $U$ has infinitely many connected components, and there is a nonzero covariant constant 1-form in infinitely many of those path components, then the space of covariant constant 1-forms will have infinite dimension. If $U$ has a finite number of path components then the dimension of the space of covariant constant 1-forms is at most the product of the dimension of $U$ and the number of its path components. In case the connection is flat, the holonomy group is discrete, but might still be very complicated, so you can't guarantee that there are any nonzero covariant constant 1-forms.