We report on time-domain stability of the parametric synchronization in a spin-torque nano-oscillator (STNO) based on a magnetic tunnel junction. Time-domain measurements of the instantaneous frequency (f(i)) of a parametrically synchronized STNO showrandom short-term unlocking of the STNO signal for low injected radio-frequency (RF) power, which cannot be revealed in time-averaged frequency domain measurements. Macrospin simulations reproduce the experimental results and reveal that the random unlocking during synchronization is driven by thermal fluctuations. We show that by using a high injected RF power, random unlocking of the STNO can be avoided. However, a perfect synchronization characterized by complete suppression of phase noise, so-called phase noise squeezing, can be obtained only at a significantly higher RF power. Our macrospin simulations suggest that a lower temperature and a higher positive ratio of the fieldlike torque to the spin transfer torque reduce the threshold RF power required for phase noise squeezing under parametric synchronization.