Let $G=(V,E,w)$ be a weighted undirected graph with $n$ vertices and $m$ edges, and fix a set of $s$ sources $S\subseteq V$. We study the problem of computing {\em almost shortest paths} (ASP) for all pairs in $S \times V$ in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of $1+ε$, for an arbitrarily small constant $ε> 0$ . In this regime existing centralized algorithms require $Ω(\min\{|E|s,n^ω\})$ time, where $ω< 2.372$ is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work $Ω(\min\{|E|s,n^ω\})$. Our centralized algorithm has running time $O((m+ ns)n^ρ)$, and its PRAM counterpart has polylogarithmic depth and work $O((m + ns)n^ρ)$, for an arbitrarily small constant $ρ> 0$. For a pair $(s,v) \in S\times V$, it provides a path of length $\hat{d}(s,v)$ that satisfies $\hat{d}(s,v) \le (1+ε)d_G(s,v) + β\cdot W(s,v)$, where $W(s,v)$ is the weight of the heaviest edge on some shortest $s-v$ path. Hence our additive term depends linearly on a {\em local} maximum edge weight, as opposed to the global maximum edge weight in previous works. Finally, our $β= (1/ρ)^{O(1/ρ)}$. We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a parameter $κ= 1,2,\ldots$, this algorithm provides for {\em unweighted} graphs a purely additive approximation of $2(κ-1)$ for {\em all pairs shortest paths} (APASP) in time $\tilde{O}(n^{2+1/κ})$. Within the same running time, our algorithm for {\em weighted} graphs provides a purely additive error of $2(κ- 1) W(u,v)$, for every vertex pair $(u,v) \in {V \choose 2}$, with $W(u,v)$ defined as above. On the way to these results we devise a suit of novel constructions of spanners, emulators and hopsets.