We investigate dissipative superradiant phase transitions and multicritical effects in an open dual-coupling Jaynes-Cummings model, where two cavity fields are simultaneously coupled to a single atom. The cavity fields are squeezed via a nondegenerate optical parametric amplification process. We show that the interplay between interactions, two-mode squeezing, and dissipation leads to rich critical behaviors, including both first-order and second-order superradiant phase transitions. The critical points are governed by the detunings between atomic (or photonic) frequencies and external pump field. Thus, these transitions can occur in the strong-coupling regime and can be easily controlled by tuning the external pump field. Utilizing the mean-field theory that explicitly neglects quantum fluctuations, we determine the phase-transition boundaries in the steady state. Additionally, we employ a full quantum description of the lowest-energy Hamiltonian in the large atomic detuning limit to analyze the quantum fluctuations. Notably, the phase diagram features two tricritical points where the critical lines of first-order and second-order phase transitions meet. Furthermore, when the two coupling pathways have specific coupling strengths, increasing the external pump strength can drive the system from the normal phase to the superradiant phase. Our work offers new insights into dissipative phase transitions and multicriticality, with potential applications in quantum sensing and information processing.

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