The Lovász Local Lemma (LLL) is a probabilistic tool which shows that, if a collection of "bad" events $\mathcal B$ in a probability space are not too likely and not too interdependent, then there is a positive probability that no bad-events in $\mathcal B$ occur. Moser & Tardos (2010) gave sequential and parallel algorithms which transformed most applications of the variable-assignment LLL into efficient algorithms. A framework of Harvey & Vondrák (2015) based on "resampling oracles" extended this to general sequential algorithms for other probability spaces satisfying the Lopsided Lovász Local Lemma (LLLL). We describe a new structural property which holds for all 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 is the key to achieving a unified parallel LLLL algorithm, which is faster than previous, problem-specific algorithms of Harris (2016) for the variable-assignment LLLL algorithm and of Harris \& 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 out of relatively simple "atomic" events. This provides the first sequential 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 $K_n$.