A permutation is called Grassmannian if it has at most one descent. In this paper, we investigate pattern avoidance and parity restrictions for such permutations. As our main result, we derive formulas for the enumeration of Grassmannian permutations that avoid a classical pattern of arbitrary size. In addition, for patterns of the form $k12\cdots(k-1)$ and $23\cdots k1$, we provide combinatorial interpretations in terms of Dyck paths, and for $35124$-avoiding Grassmannian permutations, we give an explicit bijection to certain pattern-avoiding Schröder paths. Finally, we enumerate the subsets of odd and even permutations and discuss properties of their corresponding Dyck paths.