In this paper, we mainly study two notions of pattern avoidance in parking functions. First, for any collection of length 3 patterns, we compute the number of parking functions of size $n$ that avoid them under the first notion. This is motivated by the recent work of Adeniran and Pudwell, who obtained analogous results using a second notion of pattern avoidance. Then, we provide new purely bijective proofs for two of their results, and improve the formula of another one. Finally, we apply similar enumeration techniques to the work of Novelli and Thibon on certain Hopf algebras of generalised parking functions, and compute their graded dimensions.