Question. What is the difference between a plug-in hybrid and a parallel hybrid and what would be the better choice according to you?
You're mixing terms a bit; almost every plug-in hybrid (PHEV) released in the US is a parallel hybrid. I think the question you're trying to ask is about a conventional hybrid or HEV (e.g. the original Prius) vs. a PHEV (e.g. a Chevy Volt).
Both HEVs and PHEVs have a gas engine, an electric motor, batteries, and are capable of moving on battery power. The difference between an HEV and a PHEV is that while a PHEV can be plugged in to directly charge the battery, an HEV ultimately gets all its energy from gasoline; the only way you can refuel an HEV is to add more gasoline to the tank. The hybrid system in a HEV really just serves to improve MPG a bit, whereas in a PHEV, you can generally drive all-electric the majority of the time and use the gas engine as a backup.
Just to address the other part of your question...
Among hybrids, there are basically two ways to use the gas engine: parallel or serial. All HEVs and most PHEVs are parallel, which means that the gas engine is capable of powering the drive wheels mechanically. There are a few serial PHEVs like the BMW i3 REX; in a serial hybrid, the gas engine is not mechanically linked to the wheels, and can only function as a generator for the electric motor. Serial operation is less fuel efficient at highway speeds, which is why few PHEVs are designed that way.
And finally: if you have the ability to charge your car every night, and your daily commute is less than 20 miles each way, a PHEV could be an extremely good value for you (even more so if you have, or are considering getting, solar panels).