Starting to slap parallel thumb?
 

I've used a Flea technique hybrid thing for a long time, and it served me well. But I just can't get funky enough with it. I'm trying to learn this tune, and I can't get anywhere close to the intended feel. Zender slaps through the strings, as in he follows through to the A string when slapping E. Any advice for switching? Also, what are the benefits of learning it this way?

Many great benefits to the sound and technique is all about forearm rotation, this moves the thumb off and on the strings.
You need to check out where Stu got it from, listen to the players he listened to, Mark King from Level 42 was a world leader in this style use. In the late 70s early 80s the UK produced a lot od jazz influenced bands, and of course that spanned many hits using this style in their production.
In the USA many great players to reference from Stanley Clarke in the 70s to Les Claypole in the 00s, the choice is massive. But they all share these points in their playing..check out the links.

In the first link is Stu. Zender

http://youtu.be/SdPNk9JUqRA

In this one Mark

http://youtu.be/C-7pQJN9v58

Here is Scot, check out the 4.00m mark for the movement

http://youtu.be/W70XkdHSQPA

And here is Stu Hamm telling you to think drum rudiments not just bass theory.

http://youtu.be/7sBis8v7CgU