We present fast algorithms for approximate shortest paths in the massively parallel computation (MPC) model. We provide randomized algorithms that take $poly(\log{\log{n}})$ rounds in the near-linear memory MPC model. Our results are for unweighted undirected graphs with $n$ vertices and $m$ edges. Our first contribution is a $(1+ε)$-approximation algorithm for Single-Source Shortest Paths (SSSP) that takes $poly(\log{\log{n}})$ rounds in the near-linear MPC model, where the memory per machine is $\tilde{O}(n)$ and the total memory is $\tilde{O}(mn^ρ)$, where $ρ$ is a small constant. Our second contribution is a distance oracle that allows to approximate the distance between any pair of vertices. The distance oracle is constructed in $poly(\log{\log{n}})$ rounds and allows to query a $(1+ε)(2k-1)$-approximate distance between any pair of vertices $u$ and $v$ in $O(1)$ additional rounds. The algorithm is for the near-linear memory MPC model with total memory of size $\tilde{O}((m+n^{1+ρ})n^{1/k})$, where $ρ$ is a small constant. While our algorithms are for the near-linear MPC model, in fact they only use one machine with $\tilde{O}(n)$ memory, where the rest of machines can have sublinear memory of size $O(n^γ)$ for a small constant $γ< 1$. All previous algorithms for approximate shortest paths in the near-linear MPC model either required $Ω(\log{n})$ rounds or had an $Ω(\log{n})$ approximation. Our approach is based on fast construction of near-additive emulators, limited-scale hopsets and limited-scale distance sketches that are tailored for the MPC model. While our end-results are for the near-linear MPC model, many of the tools we construct such as hopsets and emulators are constructed in the more restricted sublinear MPC model.