The small-signal stability problem of large-scale grid-connected wind farms consisting of permanent magnet synchronous generators (PMSGs) under wind power variations is addressed in this paper, with focus placed on the DC-link voltage timescale. A stability criterion for grid-connected PMSG-based wind farms under uncertain wind speed conditions is proposed based on the Routh-Hurwitz criterion. Firstly, a Weibull distribution model for wind speed is established, along with a power characteristic model of the PMSG and a dynamic equivalent state-space model of the large-scale wind farm, in which the DC-link voltage control loop and the phase-locked loop (PLL) are incorporated. Secondly, the stability probability of the wind farm is calculated, and the influence mechanisms of wind speed distribution, DC-link voltage outer loop control parameters, and PLL control parameters on the small-signal oscillation stability of the wind farm at the DC-link voltage timescale are thoroughly investigated. Finally, the correctness of the theoretical derivation is validated by modal analysis and time-domain simulations through a case study of a large-scale grid-connected wind farm comprising 260 wind turbines distributed across three sub-wind farms. It is found that the risk of wind farm instability is increased with wind speed, while the stability probability of the wind farm is enhanced as the system's critical stable wind speed is raised. The relevant conclusions can be provided as a basis for the planning of large-scale PMSG-based wind farms.