Oh, wait, I just noticed you said Parasolid. I only tried saving as IGES and importing it back, but it almost always had rebuild issues. I just looked now and Parasolid is a different file type, I thought it's just a generic term for Step, Iges. Will try your method asap. Thanks a lot, again! Very helpful.
Export the file as a parasolid. (File, Save As, choose type Parasolid). Then Import it back and save as a new file.
Parasolid is the file type that is the most "native" to Solidworks. An analogy I use: Solidworks files are like native Photoshop files, with history and layers. A Parasolid is a bitmap that has the same exact pixels but flattened, a STEP is a png, an IGES is a jpg, a STL is a GIF.