If you were going to use the generator for a tool such as a saw or drill would you drive a ground rod every time and place you used the generator?
If the generator has receptacles on the generator and the neutral is bonded to the frame then the generator is a separately derived system and would not bond to the house grounding electrode at all.
250.34 Portable and Vehicle-Mounted Generators.
(A) Portable Generators. The frame of a portable generator shall not be required to be connected to a grounding electrode as defined in 250.52 for a system supplied by the generator under the following conditions:
(1) The generator supplies only equipment mounted on the generator, cord-and-plug-connected equipment through receptacles mounted on the generator, or both, and
(2) The normally non–current-carrying metal parts of equipment and the equipment grounding conductor terminals of the receptacles are connected to the generator frame.