So Genos follows Saitama as master and Saitama the only thing that knows for increasing a hero power is training. Can Genos do any kind of training to increase his power? Or he just can increase his power when he changes his parts?
That would depend on how much modification is done on his body. From here, it says that
Genos has an entirely mechanical body in the model of a handsome young man. His face and ears look like that of a normal human, made of an artificial skin material...
If this is true, then yes, one of his only possible options is to upgrade parts only since his body is already purely mechanical. Doing Saitama's routine would not improve his already-mechanical body.
However, he has a capacity to learn and acquire knowledge. Whatever limits him due to his mechanical body can be supplemented by a constant experience and knowledge. As they say, knowledge itself is power. Power alone is useless if used without knowledge and most of the time, enemies cannot be defeated without thinking and analyzing the situation first.
Hes a cyborg only he can do is to upgrade his parts but but but it doesn't mean that he cant upgrade himself maybe he can increase his tactical combat skills and techniques in order to get stronger
also noted that in season 1 we see that Dr. Kuseno upgrades genos parts remember that part where saitama and genos fight he tries his new Anti-Saitama Tactical Upgrade