The Lovász Local Lemma (LLL) shows that, for a collection of “bad” events B in a probability space that are not too likely and not too interdependent, there is a positive probability that no events in B occur. Moser and Tardos (2010) gave sequential and parallel algorithms that transformed most applications of the variable-assignment LLL into efficient algorithms. A framework of Harvey and Vondrák (2015) based on “resampling oracles” extended this to sequential algorithms for other probability spaces satisfying a generalization of the LLL known as the Lopsided Lovász Local Lemma (LLLL). We describe a new structural property that holds for most known resampling oracles, which we call “obliviousness.” Essentially, it means that the interaction between two bad-events B , B ′ depends only on the randomness used to resample B and not the precise state within B itself. This property has two major consequences. First, combined with a framework of Kolmogorov (2016), it leads to a unified parallel LLLL algorithm, which is faster than previous, problem-specific algorithms of Harris (2016) for the variable-assignment LLLL and of Harris and Srinivasan (2014) for permutations. This gives the first RNC algorithms for rainbow perfect matchings and rainbow Hamiltonian cycles of K n . Second, this property allows us to build LLLL probability spaces from simpler “atomic” events. This gives the first resampling oracle for rainbow perfect matchings on the complete s -uniform hypergraph K n ( s ) and the first commutative resampling oracle for Hamiltonian cycles of K n .