I was careful to say, avoid ring finger in most cases, but not all. Esp. if you're an upright jazz player, you'll be doing a lot of arpeggios. Ring finger fingerings are inevitable with a lot of chording, but unlike guitar, you need to be mindful about the efficiency of ring finger use, and avoid unhealthy stretches.
I also mentioned that parallel thumb is for "most" cases, but not all. Especially up towards the first 5 frets, it's also inevitable to have to drop the wrist and pivot the thumb.
The point is, proper bass technique turns your hand into more of a mitten than a claw, and consequently, you don't have the same restrictions you do while tickling a guitar fretboard. Bass comes with its own challenges and limitations, but my original point was, with proper fretting technique, neck thickness and nut width matter less on bass than it does on guitar.