Massive stars ($M > 8$ \msun) typically form in parsec-scale molecular clumps that collapse and fragment, leading to the birth of a cluster of stellar objects. We investigate the role of magnetic fields in this process through dust polarization at 870 $\mu$m obtained with the Submillimeter Array (SMA). The SMA observations reveal polarization at scales of $\lsim$ 0.1 pc. The polarization pattern in these objects ranges from ordered hour-glass configurations to more chaotic distributions. By comparing the SMA data with the single dish data at parsec scales, we found that magnetic fields at dense core scales are either aligned within $40^\circ$ of or perpendicular to the parsec-scale magnetic fields. This finding indicates that magnetic fields play an important role during the collapse and fragmentation of massive molecular clumps and the formation of dense cores. We further compare magnetic fields in dense cores with the major axis of molecular outflows. Despite a limited number of outflows, we found that the outflow axis appears to be randomly oriented with respect to the magnetic field in the core. This result suggests that at the scale of accretion disks ($\lsim 10^3$ AU), angular momentum and dynamic interactions possibly due to close binary or multiple systems dominate over magnetic fields. With this unprecedentedly large sample massive clumps, we argue on a statistical basis that magnetic fields play an important role during the formation of dense cores at spatial scale of 0.01 - 0.1 pc in the context of massive star and cluster star formation.Read less <