Vector-symbolic architectures (VSAs) express data structures with an arbitrary complexity and perform symbolic computations on them by exploiting high-dimensional distributed representations and associated key operations. VSAs typically use dense random vectors, aka hypervectors, to represent atomic symbols that can be combined into compound symbols by multiplicative binding and additive superposition operators. For instance, a VSA-based neural encoder can bind two atomic symbols, and further superpose a set of such bound symbols---all by distributed vectors that have the same dimension. Nevertheless, decoding such an additive-multiplicative vector, to the atomic symbols from which it is built, is not a trivial task. Recently, a solution based on resonator networks was proposed to iteratively factorize one of the bound symbols. After finding the factorization, it is explained away by subtracting it from superposition. This explaining away, however, causes noise amplification that limits the number of symbols that can be reliably decoded in large problem sizes. Here, we present novel methods that efficiently decode VSA-based data structures consisting of multiplicative binding and additive superposition of symbols. We expand the pure sequential explaining away approach by performing multiple decodings in parallel using a dedicated query sampler. Compared to the baseline resonator network, this mix of sequential and parallel decoding retrieves up to 8x more additive components from larger problems in synthetic and real-world experiments.