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Our symmetry-free model for spectrum allocation (SA) in networks of general topology leverages two properties: (1) SA is equivalent to a connection permutation problem, and (2) in assigning spectrum, it is sufficient to consider the allocation made by the first-fit (FF) algorithm. This model opens up algorithmic approaches that altogether sidestep spectrum symmetry, i.e., eliminate from consideration the exponential number of equivalent solutions resulting from spectrum slot permutations. Recursive FF (RFF) is such an algorithm; it applies FF recursively to search the connection permutation space and solve the SA problem optimally. Moreover, parallelism is inherent in the spectrum symmetry-free model, as the connection permutation space may be naturally decomposed into non-overlapping subsets that can be searched independently. Accordingly, RFF admits multi-threaded implementations that may be tailored to the computing environment at hand. In this work, we present two strategies for parallelizing the execution of RFF, and we evaluate them experimentally using a comprehensive set of metrics. Our experiments indicate that RFF explores a vast number of symmetry-free solutions, and for moderate-sized networks, it takes mere seconds to yield solutions that are either optimal or very close to the lower bound.