Homomorphic secret sharing (HSS) for a function f allows input parties to distribute shares for their private inputs and then locally compute output shares from which the value of f is recovered. HSS can be directly used to obtain a two-round multiparty computation (MPC) protocol for possibly non-threshold adversary structures whose communication complexity is independent of the size of f. In this paper, we propose two constructions of HSS schemes supporting parallel evaluation of a single low-degree polynomial and tolerating multipartite and general adversary structures. Our multipartite scheme tolerates a wider class of adversary structures than the previous multipartite one in the particular case of a single evaluation and has exponentially smaller share size than the general construction. While restricting the range of tolerable adversary structures (but still applicable to non-threshold ones), our schemes perform ℓ parallel evaluations with communication complexity approximately ℓ/ log ℓ times smaller than simply using ℓ independent instances. We also formalize two classes of adversary structures taking into account real-world situations to which the previous threshold schemes are inapplicable. Our schemes then perform O(m) parallel evaluations with almost the same communication cost as a single evaluation, where m is the number of parties.