The Lovász Local Lemma (LLL) is a cornerstone principle in the probabilistic method of combinatorics, and a seminal algorithm of Moser & Tardos (2010) provides an efficient randomized algorithm to implement it. This can be parallelized to give an algorithm that uses polynomially many processors and runs in $O(\log^3 n)$ time on an EREW PRAM, stemming from $O(\log n)$ adaptive computations of a maximal independent set (MIS). Chung et al. (2014) developed faster local and parallel algorithms, potentially running in time $O(\log^2 n)$, but these algorithms require more stringent conditions than the LLL. We give a new parallel algorithm that works under essentially the same conditions as the original algorithm of Moser & Tardos but uses only a single MIS computation, thus running in $O(\log^2 n)$ time on an EREW PRAM. This can be derandomized to give an NC algorithm running in time $O(\log^2 n)$ as well, speeding up a previous NC LLL algorithm of Chandrasekaran et al. (2013). We also provide improved and tighter bounds on the run-times of the sequential and parallel resampling-based algorithms originally developed by Moser & Tardos. These apply to any problem instance in which the tighter Shearer LLL criterion is satisfied.