I assume you are talking about a series tuned (quasi) 6th order, as that's what most people use, so I'll speak from there. I've never built a parallel tuned 6th.
The issue with them being larger lies in the rear (or inner) vented chamber. It has to be quite a bit bigger than a sealed chamber to get the proper tuning. It's the part that dictate low frequency cutoff (well, kinda), so it needs to be tuned really low. From that point there's several ways you can build the "front chamber, it's not as important to have a huge ratio like a 4th. In fact some people use a smaller front chamber than rear.
To see how big your rear chamber would need to be, remember that it needs to be tuned low (probably high to mid 20hz) and you can keep the port area really small. 5-8 in^2/ft^3 is what most people use.
In my experience, a 4th order with a large ratio will get louder than a smallish 6th order with a closer ratio.