The designs of modern container ships, roll-on–roll-off vessels and cruise vessels have evolved over the years, and in recent times, some of them have been observed to experience dynamic instabilities during operation in the open ocean. These catastrophic events demonstrate that satisfying prescriptive stability rules set forth by International Maritime Organization (IMO), national authorities (e.g., Coast Guard) and other classification societies are not sufficient to ensure dynamic stability of ships at sea. In light of these events, IMO is organizing efforts to make way toward a second generation of intact stability criteria that are better equipped to deal with these dynamic instabilities. This paper discusses the development of such a tool for parametric rolling in a realistic random seaway, which is one of the critical phenomena identified by IMO. In this study, a previously developed analytical model for roll restoring moment, which was found to be effective in modeling the problem of parametric roll, is analyzed using the Melnikov approach. The stability of the system is quantified in terms of rate of phase space flux of the system. This approach is further compared with another technique known as the Markov approach that is based on stochastic averaging and quantifies stability in terms of mean first passage time. The sensitivity of both of these metrics to environmental parameters is investigated. Finally, the nature of random response is analyzed using Lyapunov exponents to determine whether the vessel exhibits any chaotic dynamics. © 2019, Springer Nature B.V.