A class of paraxial beams which are named spinning parabolic cylindrical beams (SPCBs) are introduced in free space. The SPCBs have an inherent initial momentum. The transverse position of the vortex can be controlled by varying the distribution factor, and the vortices are found to merge, annihilate, or move. The SPCBs can be regarded as the superposition of a series of half-integral charge mode and integer charge mode Laguerre-Gaussian beams. The half-integral charge mode and integer charge mode Laguerre-Gaussian beams, and the complex variable function Gaussian beams are the special cases of the SPCBs.