No. The Grounding Electrode System wire goes to from the main disconnect to the grounding rods. A ground rod can be a water pipe (if using a water pipe is legal for your house). It must be continuous with no splices; check with your local electrical supply for special welding apparatus or hydraulic crimp you can use if you're hellbound to make a splice, but the cost of the tool will be at least 10 times the cost of a wire.
Any auxiliary bonding also goes to the main panel/disconnect. I am not sure if it can have splices.
The GES wire does not go to the subpanel. The subpanel derives ground via the non-flexible EMT, RMC or IMC metal conduit between it and the main disconnect/panel. If metal conduit was not used, you must run a ground wire of appropriate rating (#6Cu or #4AL if I recall).