Three-dimensional (3D) printing continuous carbon fiber-reinforced polylactic acid (PLA) composites offer excellent tensile mechanical properties. The present study aimed to research the effect of process parameters on the tensile mechanical properties of 3D printing composite specimens through a series of mechanical experiments. The main printing parameters, including layer height, extrusion width, printing temperature, and printing speed are changed to manufacture specimens based on the modified fused filament fabrication 3D printer, and the tensile mechanical properties of 3D printing continuous carbon fiber-reinforced PLA composites are presented. By comparing the outcomes of experiments, the results show that relative fiber content has a significant impact on mechanical properties and the ratio of carbon fibers in composites is influenced by layer height and extrusion width. The tensile mechanical properties of continuous carbon fiber-reinforced composites gradually decrease with an increase of layer height and extrusion width. In addition, printing temperature and speed also affect the fiber matrix interface, i.e., tensile mechanical properties increase as the printing temperature rises, while the tensile mechanical properties decrease when the printing speed increases. Furthermore, the strengthening mechanism on the tensile mechanical properties is that external loads subjected to the components can be transferred to the carbon fibers through the fiber-matrix interface. Additionally, SEM images suggest that the main weakness of continuous carbon fiber-reinforced 3D printing composites exists in the fiber-matrix interface, and the main failure is the pull-out of the fiber caused by the interface destruction.