The distance sensitivity oracle (DSO) problem asks us to preprocess a given graph $G=(V,E)$ in order to answer queries of the form $d(x,y,e)$, which denotes the shortest path distance in $G$ from vertex $x$ to vertex $y$ when edge $e$ is removed. This is an important problem for network communication, and it has been extensively studied in the sequential settingand recently in the distributed CONGEST model. However, no prior DSO results tailored to the parallel setting were known. We present the first PRAM algorithms to construct DSOs in directed weighted graphs, that can answer a query in $O(1)$ time with a single processor after preprocessing. We also present the first work-optimal PRAM algorithms for other graph problems that belong to the sequential $\tilde{O}(mn)$ fine-grained complexity class: Replacement Paths, Second Simple Shortest Path, All Pairs Second Simple Shortest Paths and Minimum Weight Cycle.