The parallel switch sets the batteries in a parallel wiring position which is different from series wiring and produces more power . When you wire batteries in series vs parallel the power output is of one battery where parallel is a multiplication of the battery power....I think I have that right...
It's a similar concept with a two stage pump in "series" (pressure mode) or parallel (volume mode) configuration, power vs amount of output
I'm not the smartest guy around so others probably understand this better than me.
Let me give this a shot:
First, you do NOT want to connect your Yamaha Jet Boat batteries in series - the voltages will add, you'll get 24v in your electrical system and start letting the magic smoke out of things that are not designed for it.
Series means the positive of Battery A connected to the negative of Battery B. You then power your system by using the positive of B and the negative of A. Don't do this unless you are intentionally converting your electrical system (including the engine computer and starters) to 24v. In other words, don't do it on these boats.
When the "parallel" switch is open (off) then the batteries are isolated from each other (Technically, the negatives are still tied together but for our purposes they are isolated.) They are not in series (which you didn't say, but I felt it was implied) they are simply isolated.
When the "parallel" switch is closed (on) then the batteries are tied together in parallel. This means that the Positive of Battery A is connected to the Positive of Battery B and the negative of Battery A is connected to the negative of Battery B. This ADDs the available current from both batteries together. (Output power DOES multiply since power is proportional to the square of the current. "Twinkle Twinkle Little Star, Power Equals I-Squared R")
So if each battery is capable of giving you - let's say for the sake of keeping it simple - 100 amps, then in parallel they will be able to produce 200 amps working together.
So why do this? Mostly, as mentioned, if one battery is dead and you need to get power to the system that the dead battery is responsible for - usually starting the boat. (But if music is going to give you a better chance of obtaining your romantic objectives than starting the boat will, just remember you've got a pack of signal flares on-board.)
Keep in mind that if the "bad" battery has shorted internally (common on boats due to the jostling around they get), then hooking the batteries in parallel is likely to quickly drain the good battery as well.