In a capacitated directed graph, it is known that the set of all min-cuts forms a distributive lattice [1], [2]. Here, we describe this lattice as a regular predicate whose forbidden elements can be advanced in constant parallel time after precomputing a max-flow, so as to obtain parallel algorithms for min-cut problems with additional constraints encoded by lattice-linear predicates [3]. Some nice algorithmic applications follow. First, we use these methods to compute the irreducibles of the sublattice of min-cuts satisfying a regular predicate. By Birkhoff's theorem [4] this gives a succinct representation of such cuts, and so we also obtain a general algorithm for enumerating this sublattice. Finally, though we prove computing min-cuts satisfying additional constraints is NP-hard in general, we use poset slicing [5], [6] for exact algorithms with constraints not necessarily encoded by lattice-linear predicates) with better complexity than exhaustive search. We also introduce $k$-transition predicates and strong advancement for improved complexity analyses of lattice-linear predicate algorithms in parallel settings, which is of independent interest.