Notoriously I don’t believe in the influence of material on the sound of woodwinds if the internal geometry isn’t altered
But materials (especially if they are so different as wood and metal) do certainly influence the way a musical instruments built with one of these would be necessarily (due to its physical nature) built.
So a wooden flute has, as a sure difference, at the very least (because the way the walls are, if they are smooth or nor has also some degree influence) undercut toneholes while most metal flutes have raised (pulled or soldered) toneholes.
This does result in an essential difference at a crucial constructional level.
To a certain extent, this might (I am not unequivocally sure) be happening in the making of a saxophone's neck too.
Also, a neck is an integral part of continuous system. The oral cavity, the mouthpiece and reed, the neck and the saxophone are all linked together.
The shapen of the neck, its connection to the body of the saxophone AND the position of the top octave key cannot be altered without altering (positively or negatively) the whole system.
In other words the neck is not sounding the way it sounds because of the material is made of but it sounds the way it sounds because the shape it has which, in return, might be defined by the material it is made of. The connection of a new, different in shape neck, will alter some parts of the sound emission.
So, it is possible that a wooden neck has some obligatory constructional elements which make its internal geometry different.
Paraschos necks are definitely very beautiful and MIGHT have some sort of influence (positive or negative) to your sound production .
I have met the Paraschos family at the Frankfurter Musikmesse, they are very nice people and very dedicated to their products.
They told me the tale of how Yanagisawa (whom they are importers of in Greece) asked them to make wooden necks for them and how the whole thing developed.
I have tried their necks albeit not on my horn, so I cannot tell you how they changed the sound of a saxophone that I know well.
My advise is to try a Paraschos neck on your specific horn and hear what happens. Again, a neck isn’t sounding the way it does because the material is made of but because the shape it has. So there is no “ wooden” sound.
There is a sound which MIGHT be , matching your system ..............or not.
I have one for my vi tenor. There is a noted difference in the sound. I have no firm opinion as to the role material plays in the difference. What do I notice? I note a very slight reduction in power...this I find is offset by the increase in focus and that it tends to refine the tone a bit. My horn is a 64 so it has a bit more guts and power than earlier models. The paraschos seems to knock a little bite off the upper frequencies but its definitely not a tone killer. I have not had it long enough to say for sure but I think it's a keeper.
Sorry, I can't offer much help on the Bari side. I generally don't play enough or even have good enough horn to consider Bari necks. I just chimed in because I know there is little out there on these. I got my neck used. I'd be hard pressed to pay e new price due to the material and possible long term issues. I do enjoy the product though.