Airlines vary.
At my first, F/Os, even 2 stripers in BAe 146s, were allowed to talk to ground crew, start four (non FADEC) engines, taxi to the runway and take off. After landing, F/O kept control and taxiied in, and this included operations in LHR and CDG. They only handed to Captain if the stand guidance was set up for the LHS.
Next airline which flew medium and big Airbus FBW, F/Os were not allowed to taxi, despite the modern aircraft being configured for either side to taxi as standard. The pilot bosses clearly did not trust the F/Os, but they eventually gave in and allowed F/Os to taxi, but probably only because the manufacturer’s SOPs that were adopted specified this. But they still didn’t allow F/Os to taxi onto stand even if the guidance was valid for both sides.
Next airline, also medium and big Airbus FBW, allowed F/Os to do all, including turning onto stand and stopping in the right place.
However, always Captain’s hands on thrust levers after setting T/O thrust, which is fair enough, and as vilas states, SOPs are designed around this.