We propose and demonstrate an optical single sideband (OSSB) modulation approach with continuously tunable optical carrier-to-sideband ratio (OCSR) theoretically and experimentally. In the proposal, one dual-parallel Mach-Zehnder modulator (DP-MZM) acts as a key component. By properly setting the modulator, three separate sub-modulators inside the DP-MZM can be used to realize the OSSB modulation, optical carrier phase-shift, and lightwave interference. By adjusting the bias voltage of one sub-modulator, the OCSR can be tuned continuously. In the experiment, the tuning range of OCSR is found to be between-20.8 dB and 23.5 dB at modulation index m=0.2. We also analyze the relationship between the OCSR and RF power after detection. It is found that with properly adjusting the OCSR, the receiver sensitivity can be greatly improved.