How does 3 phase AC generate more voltage than single phase? Aren't the wires connected in parallel?
I think you have several misunderstandings about how AC power and three-phase AC power work.

First of all, your diagram looks like you've drawn each 'phase' with a two-wire plug, each with a separate hot and neutral like a normal single-phase AC plug. That's not how it works -- there's just three wires, one for each phase, and an optional neutral wire. (I'm ignoring the extra 'ground' wire, which exists for safety only and does not participate in normal functionality.)

It also seems like you're not thinking clearly about how AC current works, when you say 'the sum of the phases is 0 volts'. This diagram illustrates what's happening in a three-phase circuit -- the point where three wires meet on the left is the generator, or the source of the power, and the point where they meet on the right is the load, or the device using the power.



And here's an image that shows the relative voltages of the three phases over time:



As you can see, at any given moment, there is always nonzero voltage between at least some of the phases, which varies over time, and as a result of this there is always current flowing between at least two of the phases at any given time. 

The three phases are neither in series nor in parallel, because one phase wire by itself is not a voltage source in the sense that you would see in an electronics textbook. If you take one phase wire plus the neutral wire, those two wires together make an AC voltage source, which acts just like regular single-phase AC.

If you instead take one phase wire plus another phase wire from a different phase, that is also an AC voltage source -- it supplies a voltage that looks like the difference between two of the waves in the graph above, e.g. subtract the blue line from the red one. This is again a sine-wave voltage that looks just like regular single-phase AC, although it's a little harder to visualize why.

Using all three phase wires together looks like a more complicated kind of voltage source, that can't be easily broken down into several simpler sources in series or in parallel. In practice it's most often used for driving motors, which benefit from having power that flows in the way depicted in the animation above -- such that there is always current between at least two phases, with no lull to zero in the middle of a cycle like single-phase AC. This is beneficial for creating smooth rotation.