> The more common type has two parallel drivechains linked with clutches, so you still have all the drawbacks of a conventional internal combustion engine drivechain when you're using it
I don't know about the whole world, but in both the US and Europe nearly half of the hybrids on the road are from Toyota, so unless nearly everything else is two parallel drive chains linked with clutches whatever Toyota does is the more common type.
Toyota uses a series-parallel system that works by having a planetary gear system that connects the ICE, a large electric motor, a small electric motor, and a drive shaft all together.
The planetary gear system functions as a power splitting device and a continuously variable transmission. It lets them direct power flow in a bunch of different ways. Here's a summary based on Wikipedia. (MB == the bigger battery, 12V == the regular 12V batter, ICE == the ICE engine, MG1 == the smaller electric motor, MG2 == the larger electric motor):
• Aux power: MB -> DC/DC converter -> 12V
• Charge: ICE -> MG1 -> MB
• EV drive: MB -> MG2 -> wheels
• Moderate acceleration: ICE -> wheels, ICE -> MG1 -> MG2 -> wheels
• Highway: ICE -> wheels, ICE -> MG1 -> MB
• Heavy power, such as on steep hills: ICE -> wheels, ICE -> MG1 -> MB, ICE -> MG1 -> MG2 -> wheels
• Max power: ICE -> wheels, ICE -> MG1 -> MG2 -> wheels, MB -> MG2 -> wheels
• Regenerative braking: wheels -> MG2 -> MB
• B-mode braking: Wheels -> MG2 -> MB, Wheels -> MG1 -> ICE
This is a big part of why Toyota hybrids are at the top of reliability rankings. Compared to a pure ICE they replace the clutch, the transmission, the starter motor, the alternator, the reverse gear set, and the flywheel with the planetary gear power splitting device. the two electric motors, and electronics. The power splitting device has very few movings parts--just the gears themselves, a pawl that can mechanically lock the gears when parked, and fluid pumps. The gears only move by rotating, unlike in a conventional transmission where they also change position. This makes their hybrids mechanically much simpler than a pure ICE.