From a theoretical engineering point of view, the difference between parallel and series hybrids boils down to energy storage and conversion efficiencies.
As yabob points out, in a series hybrid, the engine is decoupled from the wheels, which means the engine is free to run at its most efficient speed and load. Energy conversion from gasoline to shaft work tends to be most efficient at high engine torque levels; in a “standard” drivetrain, the engine rarely runs at high torque levels, so there’s a lot of efficiency gain to be had here.
The drawback is the conversion and storage efficiencies in the additional drivetrain elements. In a series hybrid, all the motive energy has to be converted from gasoline to shaft energy (in the engine), then to electrical energy (in a generator) then to chemical energy (in the battery) then back to electrical energy, and finally to shaft energy (in the motor). Each step decreases the energy available. How much the energy decreases depends on a lot of things, because these efficiencies are not constant; they depend heavily on speed and torque or voltage and amperage. Batteries, for example, tend not to like heavy power draws.
In addition, a series hybrid would typically require some substantial energy storage–large batteries, in other words–to store all the power the engine is producing. Exactly how large is an engineering tradeoff: a very large battery pack stores a lot of energy, likely stores it more efficiently than a smaller pack, and allows the engine to cycle on and off only occasionally. However, a large battery pack is heavy, and that requires energy to haul around.
Then, of course, there’s the additional practical considerations: Series hybrids don’t require a transmission (or if they do, it would be a simplified one), so they’re simpler in that sense.
Finally, to point out: diesel-electric locomotives, which have been around for years, are series hybrids. In addition, not all series hybrids need to be electric; hydraulic hybrids can also be series or parallel, and have different sets of advantages and disadvantages.