Parametric and index-based risk transfer mechanisms are increasingly promoted as scalable solutions to climate-related losses across insurance and capital markets. However, the rapidly expanding literature on these instruments remains fragmented across disciplinary, institutional, and regulatory domains, obscuring how risk is governed and allocated under conditions of climate non-stationarity. This study conducts a concept-driven umbrella review of peer-reviewed reviews and synthesis studies to examine how parametric risk transfer is theorized, designed, and governed across insurance and capital market regimes. Using a structured search of the Scopus database and guided by the PRISMA framework, twenty-one review studies published between 1991 and 2026 were synthesized through a thematic analysis informed by core insurance principles and governance theory. An adapted AMSTAR 2 appraisal was employed to assess confidence in the evidence base. The synthesis reveals that parametric instruments operate under fundamentally different governance logics across market regimes, despite shared reliance on index-based triggers. In insurance-centered applications, trigger design is closely tied to principles of indemnification, insurable interest, and legitimacy, with basis risk framed as a social and contractual concern. In capital-market-based structures, including catastrophe bonds and insurance-linked securities, trigger design functions primarily as a risk allocation mechanism, shifting model uncertainty, basis risk, and climate non-stationarity from sponsors to investors. Across both regimes, the increasing reliance on complex models and externally produced data reconfigures traditional insurance principles, relocating key governance functions from underwriting and claims settlement to model governance and index calibration. These findings suggest that parametric risk transfer should not be understood solely as a technical innovation but as a governance arrangement that redistributes climate risk, uncertainty, and accountability across actors. The paper contributes a unifying conceptual framework that links trigger design, insurance principles, and regulatory boundaries, highlighting implications for climate risk governance, market regulation, and the future role of insurance under accelerating climate change.