Recent studies show that the loss of stability for a voltage-source converter (VSC) in weak-grid connection is largely related to its synchronization unit, i.e., the phase-locked loop (PLL). This paper studies the synchronization stability of a system comprised by two VSCs in parallel connection to a weak grid. A reduced transfer function based small-signal model, which can allow for the interactions between PLL and converter outer power controls, is first proposed. Then, an improved net damping criterion is used to analyze the damping and stability characters of such system under various operating conditions and different controller configurations. Compared to the conventional net damping criterion, the used criterion has wider applicability in terms of stability judgment. Case studies show that the studied system tends to be unstable at weak-grid or heavy-loading conditions. The instability can be in the form of oscillations or monotonic divergence, in which, the latter is more likely to occur for the converters without grid voltage regulation capabilities. Besides, the net damping-based sensitivity studies can provide guidance on control tuning or design for stability enhancement. Detailed model-based time domain simulations are conducted to verify the analysis results.