Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low (almost-) independence. A series of papers, beginning with Luby (1993) and continuing with Berger and Rompel (1991) and Chari et al. (2000), showed that these techniques can be combined to give deterministic parallel algorithms for combinatorial optimization problems involving sums of w -juntas. We improve these algorithms through derandomized variable partitioning, reducing the processor complexity to essentially independent of w and time complexity to linear in w . As a key subroutine, we give a new algorithm to generate a probability space which can fool a given set of neighborhoods. Schulman (1992) gave an NC algorithm to do so for neighborhoods of size w ≤ O (log n ). Our new algorithm is in NC 1 , with essentially optimal time and processor complexity, when w = O (log n ); it remains in NC up to w = polylog( n ). This answers an open problem of Schulman. One major application of these algorithms is an NC algorithm for the Lovász Local Lemma. Previous NC algorithms, including the seminal algorithm of Moser and Tardos (2010) and the work of Chandrasekaran et. al (2013), required that (essentially) the bad-events could span only O (log n ) variables; we relax this to polylog( n ) variables. We use this for an NC 2 algorithm for defective vertex coloring, which works for arbitrary degree graphs.