Since 2008 June, the BU blazar group has been carrying out monthly monitoring with the Very Long Baseline Array at 43 GHz of a sample of gamma-ray blazars. We present general results on polarization parameters of the VLBI cores at 43 GHz for different sub-classes of blazars (FSRQs, LSP, ISP, and HSP BLLacs). We find that the LSP BL Lac objects possess the highest degree of polarization, while the HSPs have VLBI cores with the lowest degree of polarization. The position angle of polarization (EVPA) in the HSP BLLacs is the most stable among the blazars, with the EVPA tending to align with the jet direction. The FSRQs exhibit the highest variability of polarization parameters. We explain findings within a model of the magnetic field in the parsec-scale jets consisting of different contributions of ordered and turbulent components. We compare the behavior of polarization parameters with gamma-ray activity of the blazars. This research is supported by NASA Fermi Guest Investigator Program grant 80NSSC17K0649 and NSF grant AST-1615796.