Okay Cowboy, I gotcha!
Parabolic reflectors have been around since, well shit, they've been around forever! They have a very loyal following, so they they must work fine. I've never used one though, so I really can't say much about them.
Now, the "inline, four-sided, reflector" (that sounds so formal!). Most people just call them air cooled hoods, or air coolable hoods. There are two main advantages here:
1. By adding a fan and ducting directly to the hood, you can remove most of the heat produced by the light - allowing the hood to be placed much closer to the plant. DON'T UNDERESTIMATE THIS ADVANTAGE! When light has to travel twice as far it only delivers 1/4 of it's energy! Think about that for a minute! It's all about how close you can get your light to your plants! That's not just my opinion it's the "Inverse Square Law" - it's Physics!
2. A four-sided reflector is best able to project light into a conventional (four-sided) grow area, with minimal loss due to distribution irregularities.
Now, with that being said - a proper design makes all the difference in the world. A well designed cooltube with a batwing reflector might outperform a "so-so" air cooled hood. But when push comes to shove, give me a well designed air cooled hood every time! That's my story and I'm sticking to it!